I have a FAQ page that seemed like a useful thing back in 2004 and it has been kept up to date. It is noindexed and disallowed because I do not think it is a good place to come in to the site.
My experience concurs with the wisdom of noindexing (at least some) FAQs... albeit it was a learning process, not something that I'd anticipated.
In 2004, I think that all sites had an FAQ page, and that some SEOs came to use them as doorways. During the early days of Panda, Google was displaying query-sensitive extra SiteLink type links, extra inline links, in the SERPs... and my guess at the time (which I think was correct)... was that Google was using the extra links it displayed actually as a
culling mechanism... one of many types of serp configurations they have deployed, to provide a basis of comparison for a systematic process of elimination, to end up with the most desirable result.
The serp-crowding results, where Google offered three or four or more pages from the same domain, was another culling mechanism.
In the case of the early Panda, I remember that when a certain client got an extra inline link for its FAQ page, I was actually alarmed, because that was the page on which the client included what were essentially the fine-print caveats about things like non-returnable items. This was something I'd felt needed to be more prominent on product pages and in the shopping cart.... and we argued about it... but FAQs were where the client felt caveats worked best. As a landing page, though, that FAQ page would have been a disaster. There were also concerns that a jump-back to the serps, eg, might hurt the site's reputation with Google, and I noindexed the page. It was one of my rare uses of noindex for a page meant to be seen at all.
IMO, "People also asked" is very probably a similar selection mechanism, which also provides Google with additional statistical information, kind of a multivariate testing... and it also provides users in a hurry with a way of getting a quick answer... and of deciding whether they want to go more deeply into the site from which the answer was chosen, or return to the serps, or to exit Google. I'm thinking the process also offers Google a way of determining what the best order of also-asked questions might be for certain queries... and the fast shuffling through offerings provide for quick and easy sampling.
The "People also search" results do seem to be getting more focused over time... and I need to check some test searches I'd set up way back to see what the changes are.
I've been noticing very recently, during the pandemic, that many state and local government sites are using FAQs as a lazy way of updating frequently changing information on their sites... ie, they are
always expecting users to refer to the FAQ. They are not, though, always making this expectation explicit, and that, IMO, can lead to problems.
Perhaps users should assume that FAQs are must-reads... or sites should not assume that they are. I don't know... but the noindex might create problems for this particular way of updating information.