(Disclosure: I review for MathSciNet.)
how much of the responsibility lies on the MathSciNet reviewer who "validated" the proof?
In my view, very little if any, and by using the word "validate" I think you overstate the case. Reviewing for MathSciNet is not meant to be like refereeing, and reviewers are definitely not asked to check the correctness of a paper as they would for a referee report. The purpose of a MathSciNet review is mainly to summarize what the paper contains, so that prospective readers can quickly tell if the contents of the paper are likely to be of interest to them. I don't think you should take the existence of a MathSciNet review as any sort of "seal of approval" on the correctness of the paper.
The reviewer is certainly allowed to mention errors or shortcomings in the paper if they should happen to observe any, but this is not a requirement or an expectation.
And keep in mind that unlike an referee report, MathSciNet reviews are public to the world and signed with the reviewer's name. If a reviewer says something negative about a paper, especially a very high-profile paper, they might face blowback in a way that an anonymous referee wouldn't. Maybe you think they have an obligation to do it anyway and face the consequences, but it's a significant ask.
Useful reading is the Guide for Reviewers that MathSciNet points reviewers to.
My understanding is that it is their duty to reveal the mistakes in the published article, so that the other authors do not base their work on it.
I would only agree insofar as to say that if the reviewer is actually aware of a specific mistake in the paper, they should mention it. Even so, that would only be out of a general civic duty to the community, not any responsibility explicitly laid on them by MathSciNet. And for me, a vague sense of unease that something is not quite right with the paper would not rise to that level. If the reviewer does not happen to find a specific error, I do not think they have a responsibility to go looking for one. Certainly in my time as a reviewer and in talking to others, I've never had an impression that this was expected.
Is there a way to "nudge" the reviewer to make amends to the review?
Their name is attached to the review, so of course you are free to contact them and say something. But as I mentioned, I think what you're asking is beyond the scope of what's expected of a reviewer. If you think the reviewer actually knows of a specific error and has covered it up, that would be another story, but as I said I don't think they have an obligation to carefully check a paper in hopes of finding errors.
Moreover, MathSciNet doesn't really encourage reviewers to revise their reviews after initial submission. There is no automated system to submit revisions. So even if the reviewer wanted to make changes, they'd have to contact the MathSciNet editorial staff and convince them that changes were warranted.
Now some other articles (published in first-tier journals) use this questionable result.
Well, it's the responsibility of those authors to satisfy themselves of the correctness of results that they rely on. They cannot reasonably take the mere existence of a MathSciNet review as positive evidence of the paper's correctness.