Why is the spoken German in many US films and TV shows so inaccurate?

Because it's a level of detail that was considered irrelevant for the show's development

As a software developer and general IT nerd, I am constantly faced with this principle. Many, many TV shows and movies forgo showing a correct approach, and opted for the most minimally correct display of IT, because they assume it's a pointless detail.

• I think this is the most obvious example. You don't even need to be an IT expert to understand that this is completely ridiculous.

• From CSI: "I will create a GUI interface using visual basic to track the killer's IP address". This is technical lingo, but it is pulled out of context. This is the equivalent of a car mechanic saying "I will crank the carburator through the horsepower, so that I can clutch the shaft". The words themselves sound thematically correct, but there is no meaning to this random succession of words.

I see it happen more with IT than German (as I don't speak German), but the principle is the same: being factually correct was deemed to much of an irrelevant detail, and no one wanted to put in the effort to research the correct thing to say.

Because no one working on the movie knew German well enough to spot the mistake

Even if the script contained correct German, and the actor mistakenly messes up the grammar, it's still possible that no one catches it.

If the movie is recorded in the USA with an American crew, none of them having any German speaking skill, then no one will notice that the actor said it wrongly.

Secondly, in the case of more than just a single line of foreign language, actors often go to special trainers to teach them how to sound like a true German, even if only for a few lines.

Because of this, most of the people working on set will not correct the actor; since they know he has been trained to speak German and they know little or nothing about it. By comparison, they feel inadequate to highlight the (trained) actor's mistake.

Because it's a joke

You've mentioned How I Met Your Mother twice. In both cases, the butchering of the German language is intentional.

• In the first example, HIMYM mocks the German language for having long words that sound like gibberish but actually have a very complex and nuanced meaning. If you think that's just not true, torschlusspanik.

• In the second example, HIMYM mocks the German language again for sounding like gibberish. Heidi Klum's presence (since she is an actual German speaker) supports this joke. If an American actor had said it, the viewers could assume that the German sounded horrible because the actor was not able to speak German. But they specifically had Heidi say it, to prevent people from thinking it was an actor's shortcoming, and therefore understanding that German is "truly" a gibberish language (I say "truly" because that is the in-universe funny truth, not the out-of-universe real truth).

Edit
Following the suggestion the in comments, here are better examples of ridiculously long (existing) German words:

Rindfleischetikettierungsüberwachungsaufgabenübertragungsgesetz
Beef labeling supervision duties delegation law

Donaudampfschiffahrtselektrizitätenhauptbetriebswerkbauunterbeamtengesellschaft
Association for subordinate officials of the head office management of the Danube steamboat electrical services

There are a few potential reasons for this:

• Rule of Funny. Most of the shows you linked are comedies, and someone speaking pseudo-German nonsense in a bad accent is inherently funnier than someone speaking actual German (unless you're a native German speaker).
• It's not worth it. Again, most of the clips are from sitcoms (which are not $100m productions), and are only a single throwaway scene. If the German was pivotal to the plot, they'd be more likely to take the time and effort to make sure it was accurate, but for a single scene they can just make up whatever and say "yeah, that sounds good enough". [This is also the reason why, in many old WWII movies, the Germans all speak English in bad accents rather than German]
• They didn't think anyone would notice. These shows are made primarily for an American audience, and most Americans don't speak German. They'd watch these scenes and think the German is perfectly valid.
• Alternately, a Genius Bonus/Bilingual Bonus for anyone who does speak German. They'd watch the scene and be able to recognize, as you did, that the person is speaking nonsense.

It's worth noting that this phenomenon isn't unique to English-language works, either. Anime in particular is infamous for its terrible English (partly for budgetary reasons, partly due to linguistic differences). My personal favourite is Kiniro Mosaic, in which two of the main characters were born and raised in England and one of them is pure-blooded English, yet neither of them sounds even remotely English when they speak:

It's not just German. Pretty much every "foreign" language suffers the same fate in Hollywood movies.

(Exception: Spanish, because a large minority of Americans actually understand a little bit of Spanish. Indeed some are even native Spanish speakers.)

Consider for example Captain Fantastic (2016) which had pretensions of being an intelligent film and where everyone in the family depicted was super-intelligent and capable:

• The 8-year-old knew all about the Bill of Rights and some 2010 Supreme Court Case. And could articulate an intelligent opinion about it.
• Father was going to test son on quantum entanglement and Planck length vs Planck time.
• They all played/sang amazingly beautiful music.
• And they weren't just bookish nerds. They live outdoors, hunt their own food, do military-style exercises everyday, and can fly over rooftops ninja-style (providing there's no falling tile).

But unfortunately, there was this scene where they started speaking in a variety of "foreign" languages. The purpose of this scene was to impress the viewer with yet another one of this remarkable family's amazing talents.

But unfortunately the "foreign" languages were spoken horribly and it was obvious they had just been lines memorized from a script. So if you actually understand any of these "foreign" languages, this is one part of the movie that kinda falls apart and undermines the whole "super-intelligent hippie family" premise of the movie.

The Mandarin was particularly and comically awful. (On my first viewing, I could vaguely tell that Mandarin was probably being spoken. It was only when I was writing this answer and watched this clip several more times that I could make out what exactly the guy was saying.)

(And would a German speaker please also let me know if "Ich können Deutsch sprechen" is even grammatically correct? @TaW: "No it isn't. Correct: Ich kann Deutsch (sprechen)".)

But the thing is, most Americans (including the writers and director) understand only English and wouldn't know anyway. The writers don't care, the director doesn't care, the actors don't care, the viewers don't care. So why bother wasting even a bit of money or time on something that no one cares about?

(Some have argued in the comments below that in Captain Fantastic, "foreign" languages happened to be the family's one and only blemish among their multitude of amazing talents. But I favor a simpler explanation: the director, writers, and actors simply didn't bother getting this right or didn't know how to.)

I think though that nowadays for blockbuster Hollywood movies directed at a global audience, they do make more of an effort. (These days, for blockbuster movies, the US box office revenue is usually less than the outside-US revenue.) So for example Arrival which featured a somewhat prominent role for Chinese characters actually bothered making an effort. Indeed they even spent some good money inventing the alien language. (Though in the brief scene where Amy Adams was speaking Mandarin, it wasn't very good, which was inconsistent with her being a polyglot linguist who could speak Mandarin.)

But Captain Fantastic and older movies like Die Hard were definitely targeted only at an American audience.

Note: My answer doesn't explain the Heidi Klum bit in HIMYM which I've never seen.

All of those shows you listed were created in the USA for America by Americans. Given that, you have to realize that very, very few Americans know German.

I'm talking less than half a percent of the country (< 0.5%) . Even worse, a large number of those speakers are Amish or Mennonites, who are not very likely to go buy a ticket to a Die Hard movie.

To give you a bit of perspective, more Americans speak Tagalog and Vietnamese than speak German. The language is just not that important in the USA.

If you are a musician, you can tell that most music pictured in movies is obviously phony. If you are in the military, you can tell that most depictions of the military in the movies are obviously wrong. And so on. It only has to be convincing to most of the audience. I think probably the situation with German is neither better nor worse than most such details.

This is too long for a comment (other answers are more comprehensive), but for the HIMYM instance, this is not so much mocking German as it is mashing up English as German for comedic effetc. There is a long tradition of this because English so easily "transforms" into Germanish with a few simply tricks. I think this is quite different from the throw-away lines in the more serious contexts, especially in light of the reuse of the term as a gag.

Leben sounds like Lieben so obviously, this was chosen. Slangen is simply English slang. Schiksa is a disparaging "jewish" (Yiddish?) term, generally for a non-jewish woman and is often used winkingly as in "She's got shiksappeal". Shatz is a play on the past tense of shit (at least, informally it is). The exact spelling in the CC may be "wrong" in that often CC is by a transcriber and not from a script.

A standard example of this format of comedy can be seen in an old sign that ran the rounds in computer labs: (in blackletter, all caps) "ALLES TURISTEN UND NONTEKNISCHEN LOOKENPEEPERS! DAS KOMPUTERMASCHINE IST NICHT FÜR DER GEFINGERPOKEN [...]"

Die Hard, I would think, would probably spent a little money to get it right. It was headed for International release, so language would be important. However, you're talking about a Shakespearean actor of British descent with no experience in the language, so the fact that he missed a subtle nuance wouldn't be surprising. I mean, if it was that easy then everyone could be a spy.

I'd like to shadow something mentioned by Flater: this is the norm when dealing with IT. Go back and watch the Sandra Bullock vehicle, The Net. OMFG, they got just about everything wrong in that movie. But, who but the most hardcore of nerds would have known? I think TV shows probably operate under the same assumption about other languages; very few people would know. Well, with regards to Spanish they're probably getting more sensitive, but that's it.

This isn't just TV/movies; I'd hazard a bet that the German used in Frank Zappa's "Stick It Out" probably isn't very accurate either. And Zappa toured Europe all the time.

Considering the Die Hard scene as an example. The intent is not that you understand what he is saying but rather that you understand he is a foreign terrorist. Therefore the emphasis is on using words an American/English audience would stereotypically view as German and in a tone that American/English viewers hear as consistent with the role.

The correct native speech may lack this and therefore sound strange to the target audience.

Similar to the tech example where being factually correct sounds boring compared to stringing a few unrelated slightly recognisable buzz words together.

Even when the producers know that there is a mistake, and the mistake isn't serving the context, they permit mistakes to be made and to go uncorrected.

I've seen Hollywood productions in which Hollywood production is incorrectly portrayed.

We are talking German here. That is a language no non-native speaker can hope to speak acceptably anyway, so you cannot expect the same care to detail as given to, say, Elvish or Klingon. Speakers of High Elvish understand Wood Elvish fine. Speakers of High German will not understand Low German speakers, and neither will be able to understand or speak Letzeburgisch or Yiddish. If you learn German in school, that's good for listening to news broadcasts intended for international audience ("Deutsche Welle"). And you'll be able to make yourself understandable in several parts of German (though in some getting addressed in High German is considered insulting, they are forgiving with foreigners and other less gifted people).

So why bother? You cannot win anyway.