Why did the Wachowskis change the story's themes so significantly?

The core answer

My question is why would the only reference to anarchism in the movie be actually a disturbing reference to chaos which actually has nothing to do with anarchy?

This is not the only reference to anarchism. Everyone who wears the mask and goes out that night is an anarchist, as they have all rejected the government's rules.

What you're focusing on is violent and chaotic anarchism. You're actually striking on an important part of the interpretation of what anarchy is, compared to how it is perceived.

Anarchy is the rejection of the notion that there is a person (or group of people) which holds more power than another person (or group of people). Anarchy, at its very essence, rejects hierarchical societal structures in favor of an equal footing.
What communism is to money and property, anarchy is to power. It is the belief that wealth/power is shared by all people and not held by a select few.

However, the opponents of a political ideology will always point to the worst cases. The use confirmation bias to assume that if they prominently talk about the worst cases, that people will subconsciously start assuming that the worst cases are the most prominent cases.
For anarchy, that means that violent anarchist are claimed to be the alleged true intention of any anarchist movement. For communism, that means that Stalin's corrupt regime is claimed to be the only true intention of any communist movement. In short: while abuses definitely exist and denying that would be incorrect, it is equally incorrect to assume that therefore the ideology itself is inherently intent on abuse.

When V tells people to stand up against Sutler, they (eventually) do so. V gives them the mask as a way for them to rise up against the establishment but avoid personal persecution for doing so.
However, just like any tool, the mask can be used for good or for evil. If you give people knives, some will become robbers, others will cut their steak, and others become surgeons. The masked robbers are using it for evil (crime), whereas the masked march uses it for good (peaceful protest).

The scene with the masked robbers showcases that Sutler's point is not fully incorrect. Sutler argues that personal freedom leads to abuse of personal freedom, which leads to crime and other negatives to society. And while this is true for specific cases (i.e. the masked robbers), Sutler very much omits everyone else who does not use their personal freedom in the pursuit of abusing others.

This is something you can find in almost all political discourse of the last century. It is why the US still uses "communist" as an irrevocably bad idea (because they measure it by the worst of the worst, i.e. the corruption under Stalin's communism), or the argument that gun ownership "must" signal criminal intent, or that anyone who takes/smokes drugs is "obviously" helplessly addicted.

I'm not staking any political claim here. The same rhetoric happens on all sides, regardless of political leaning. I'm just pointing out that it's very relevant for V for Vendetta to show that there will always be some people who abuse a system, and this is true for any system. There were abuses in Sutler's regime, and V's masks have led to some people abusing them (by committing crimes).

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Your questions

Do the Wachowskis have a point with that?

See the above.

Did they deliberately strip off anarchy from the movie?

They did not. You simply misidentified what is and isn't anarchy.

Is it an American Culture thing?

The movie, compared to the comics, are definitely made more relatable to an American political context. But the gist of the story remains the same.

Can it be following a political idea?

V is still a vocal proponent of anarchy. The distribution of the masks very much puts people on equal footing, as opposed to e.g. uniforms that denote rank (and therefore hierarchical power).

Or is it just that the Wachowskis weren't sure anarchy was a good thing so they stripped it off, but if so why did they actually somehow sabotages anarchy instead of being neutral to it?

As mentioned before, they did not strip off the anarchy. They simply didn't explicitly point out that the silent march was also a form of anarchy.