This answer is based on my comment exchange with Schrödinger's cat below the question Request to undelete.
User requests for a specific question serve two different purposes: first to educate users by explaining why their question answer was indeed worthy of closing or deleting, usually paired with downvotes on the meta question, and second in some cases to correct a wrong decision by reviewers.
Valid answers consist of an explanation why something was closed, which can be accepted if the OP understands the reason, or by explaning (or guessing) why a review mistake has been made, which can also be accepted by the OP. Votes are used to express agreement or disagreement with the closure. It's useful to have such things on Meta, because other new users can (ideally) learn how the site works by reading previous discussions on specific questions.
So, to address the questions raised in this post:
- Can one answer such questions, if so how?
One can answer a reopen/undelete request by explaining why a post was closed/deleted, in more detail than through an automatic close comment. If this explanation is very similar to earlier cases then you can also close the meta question as a duplicate of another specific or canonical meta post explaining the reason.
One can also answer by acknowledging that the closure/deletion was unjustified or questionable, or start/respond to a discussion if this is indeed the case for the particular question being discussed. An answer of this type can also state that a reopen/undelete process is started (or completed).
- Who can answer such questions?
Regulars on the site can answer most of these questions. For some of them the input of the specific reviewers or moderators involved in closing/deleting may be required.
- If an answer is not possible, how is this a question?
Meta has many different functions, for example company announcements such as the yearly summary, or discussion topics in the discussion tag. These "questions" do not necessarily fit in the Q&A format, in terms of being answerable, how much sense it makes to accept an answer, or what the interpretation of votes on the question and answers is. For Meta that is not really a problem, as long as it serves a purpose. For the current category of specific reopen/undelete requests one of the functions is to allow people to engage with the community about their specific question or their understanding of the rules, as applied to the concrete example of their current question. Even if that is not answerable, or not useful for anybody besides themselves (which often is not the case, i.e., such a request is answerable, and useful for the community in terms of teaching by example) then such an interaction is still valuable in terms of communication and participation, or giving people a platform to discuss their question. An alternative is chat, which is indeed used for this purpose as well, but Meta has advantages over chat such as being more contained, more persistent, better presentation of information, better voting mechanisms, better visibility/discoverability.
- Is the idea to then write an answer saying it was undeleted?
- What happens if it does not get undeleted within some amount of time?
A request may be unresolved, with or without a meaningful discussion. Then the system has failed, in a way. This happens on main as well.
- What is the benefit of having such posts around for many years?
It is an archive of site policies and policy discussions, applied to example questions. This is useful for regulars to think about their own interpretation of the rules. It is also useful to be able te refer new users to previous examples, as a sort of case law. And it is useful for new users to educate themselves, i.e., their question is closed, they go to meta to complain, they start writing a meta question, related questions are suggested, they read the related questions, they decide not to post their request (ideally).
- Why don't we have tons of such posts from the past?
On TeX.SE meta is not used much for anything, many people (especially the people who would like to have their question reopened) may not know that meta exists, or that this kind of questions can be asked. That doesn't mean that it is "off-topic" or not useful on meta. Note that the meta at Stack Overflow has a specific tag for this exact purpose: reopen-closed with as tag guidance "A request to have a specific question reopened by the community, often resulting in explanations why it was closed and guidance on improving it". Many of those requests are downvoted. This does not mean the requests do not belong on meta, just that people disagree with reopening the question.
- When should one use the undelete and reopen buttons and when to ask on meta?
Use meta for explaining why a post should be reopened or undeleted, or to answer why you agree or disagree with the arguments presented. Use the buttons to actually reopen or undelete a post if the meta discussion provides reasons of doing that.
reopenbutton is the way in which things actually get reopened. But it simply casts a vote for that action. And since close voting can happen for the wrong reasons (we have many meta posts on this phenomenon) I see no problem with using meta to raise issues with a question you believe is wrongly closed.