Is the "people expecting to die don't lie" catch really admissible in court?

"people expecting to die are less likely to lie so it is admissible in court

It is admissible in court. A dying declaration is defined as follows:

In the law of evidence, a dying declaration is testimony that would normally be barred as hearsay but may in common law nonetheless be admitted as evidence in criminal law trials because it constituted the last words of a dying person. The rationale, accurate or not, is that someone who is dying or believes death to be imminent would have less incentive to fabricate testimony, and as such, the hearsay statement carries with it some reliability.

However, a dying declaration is not assumed to be true. It's more a matter of not being as imprecise as hearsay.

Hearsay is inadmissible because it can't reasonably be proven. If I tell you that Frankie told me that Tommy confessed to killing Frankie's brother, there are many steps needed to verify this account:

• Am I telling the truth? Did Frankie tell me that? The odds of me conclusively proving that are slim to none.
• Even if we prove it, was Frankie telling the truth? Maybe he was lying to me. How would I go about conclusively proving Frankie's intentions? Even if he wasn't lying, how could I prove that Frankie wasn't just making unwarranted inferences, speaking figuratively, or simply exaggerating?
• Even if we prove that, how do we know that Tommy was being truthful to Frankie? Maybe Tommy was spinning a story, or exaggerating, or using a figure of speech, or ...

It is nigh impossible to ever verify a hearsay account because there are so many reasons for it to not be true. A very simple example is someone who lies and boasts - they may have claimed to have done something, but that doesn't men they actually did it.
Hearsay is information passed through the "truth filters" of many people, and thus not direct testimony.

However, a dying declaration is admissible, if the declaration is given directly by the dying person (as is the case here, through the video) or by having one level of "thruth filter".
You can testify what your dying brother told you. You cannot testify what Tommy told you about what Tommy's dying brother said to Tommy, as that again becomes hearsay.

Now understanding the reason for why hearsay is indamissible, the conclusion to your question is that a dying declaration is not assumed to be true, but the (now dead) witness is unable to be questioned directly and the lowered likelihood of a dying witness lying for personal gain sufficiently alleviates the otherwise unverifiable nature of hearsay.

Yes, at least in the US and the UK. It's called a dying declaration, but the point of it is that it's an exception to the usual rules excluding hearsay evidence and requiring witnesses to be available for cross-examination. In this case the testimony has already been brought before a jury, so it would be pointless.