Addressing the question at the end of your post, the FAQ is intentionally left vague because (a) the guidelines are, to an extent, network-wide boilerplate without a lot of room for per-site customization, (b) the FAQ is not easily changed, and (c) community standards are meant to evolve over time and the FAQ is not meant to interfere with shifts in scope and emphasis. More detailed guidelines, policies, and procedures are ironed out via meta discussion. This is by design.
Now, addressing your purported contradictions point-by-point:
The job of a peer-reviewer is to scrutinize a paper and check the details. It typically takes me many hours, spread out over a week or two, to peer-review a paper for a journal (others may work faster or slower---I can only describe what is typical for my own workflow). Peer-reviewing a paper is (or should be) an intense activity. Math SE is not setup to provide this kind of scrutiny, even if a paper is so deeply flawed that a potential answerer can spot fatal errors in the opening sentence.
As a general policy, requests for peer-review are far too broad for Math SE, and are, therefore, off-topic.
The FAQ: "Your questions should be reasonably scoped. If you can imagine an entire book that answers your question, you’re asking too much."
Xander: "Math SE is intended to be a repository of narrow, focused questions with high-quality, authoritative answers"
Again, I see no contradiction here. Questions should be focused. The FAQ gives an extreme example---a question which requires a book to answer to way too broad. However, a question can be too broad but not require a book to answer.
Generally speaking, questions which present an argument then demand that MathSE users "spot the error" are too broad. This is true even of the proof-verification tag (which, by the way, is not a tag without controversy): a good proof-verification question should discuss specific questions about the correctness of a proof, and should focus on specific uncertainties that the author may have.
From the tag-wiki:
This tag should be used when you have a proposed solution to a problem that you want to have looked over by somebody else. It is useful for these questions to include an explanation for why the argument they present is not convincing enough. Further discussion on using this tag can be found in the Mathematics Meta post here. Questions of the type "I have written this proof, is it right?" are, typically, too broad.
Also relevant is the meta thread The problem with proof verification and the links in the question. Keep in mind that policy on Math SE is meant to be arrived at collectively---perusing those threads, I come to the conclusion that the proof-verification tag is meant for questions about proofs written by the question author, where the author is seeking clarification regarding specific issues in their argument.
All of this is doubly true when the proof being verified is a claimed proof of a major open problem.
The FAQ: "There are certain subjects that, while still on-topic here, might be better addressed by one of our sister sites: Research-level mathematics ..."
Xander: "Research level questions are not, generally speaking, on-topic here."
I see no contradiction here. The FAQ states that certain topics---specifically, research level questions---are more appropriately asked on MathOverflow. The general SE model asserts that any particular question should only be asked on one site in the SE network (as a general rule cross-posting is discouraged). Ergo if a question is more appropriate on one site than another, it is out-of-scope or off-topic on the site where it is less appropriate. Therefore, as a general rule (and not as a hard-and-fast stricture), research level questions are not on-topic here, as they are on-topic on MO.
The line between the two sites is fuzzy, but, speaking in generalities, research-level questions don't belong here, and will often be migrated to MO. The goal is to find the right home for any particular question, and I don't imagine that a question would be closed on Math SE simply because it is research-level---again, migration is a more typical outcome. Of course, if a research-level question is otherwise fine, and is not welcome on MO, then it is likely that it will have a home on Math SE.
Regarding the Specific Question Discussed Above
The question https://math.stackexchange.com/q/4197244/ is off-topic on MathSE for a number of reasons. The most salient problem with the question is that it is asking the Math SE community to peer-review (to "find the bug in") a paper written by a third party. Math SE simply isn't setup to provide this service.
Taking a step back, the question is also poorly-scoped. If the question were "I have read this paper, and am confused by (47) on page 23. How does the author arrive at this identity?", then we could reasonably expect the Math SE community to weigh in. In this case, any potential answerer is expected to read the entire paper---the focus of the question is not on any particular step or argument, so the question is too broad. Thus, even if the question were not a request for peer-review, it would be off-topic for Math SE.
Finally, and this is really the smallest flaw in the question, research-level questions are typically not on-topic here. The line between Math SE and MO can be blurry, and it is not clear that a reasonably-scoped question about this particular paper would be off-topic on Math SE, but, as a general rule, questions on active research topics belong on MO, not Math SE.