(TL;DR at the end)
I'm not someone who frequents the site a lot, but I ended up here after seeing this answer, and wondering why it had been deleted. (I still don't know for sure, but it certainly does look a lot like a ChatGPT-generated answer.)
I actually thought it was a decent answer, and was at a loss to determine why it had been deleted. (Hence the investigation, and ending up here.)
I know there can definitely be concerns about the quality of content generated by ChatGPT, but there are also concerns about the quality of content generated by human users of the site. A lot of human-written posts are downvoted or deleted because they are wrong/vacuous, or because of rep farming.
I used to be a moderator on Puzzling.SE, and we had a similar (though not identical) problem where people would post puzzles they found online as a way to attempt to quickly gain recognition. There, we adopted a policy that any puzzle that was not original needed to be attributed, and puzzles from sources that did not allow copying or reposting would be deleted.
When a puzzle was suspected of being non-original, the user would be asked to provide attribution, or to assert that it was in fact original. If they did not, the post would be deleted.
Obviously, those not complying with the requirements, or repeatedly posting disallowed content after being warned, would be suspended.
I think a similar policy would be useful here. Require any ChatGPT-generated posts to be clearly labelled as such ("this answer was generated by ChatGPT" or something along those lines). If an answer is suspected of not being original, ask the user whether it was written by ChatGPT, and if so, to add a label.
There could be a policy that a user can only post x ChatGPT-generated responses per day/week/month/whatever to prevent users from rep-farming.
I think this addresses all the points mentioned above except the increase in moderation load, but I think that is unavoidable no matter what solution is chosen. The moderators will have to deal with ChatGPT-written posts no matter what policy is adopted. I suppose if it becomes overwhelming, additional moderators could be added.
TL;DR: Many AI-generated answers can be helpful (and many human-generated ones are not), so why not let them be "moderated" by voter opinion? But they should be clearly labelled as AI-generated, and users attempting to pass them off as original should be dealt with appropriately.