APA style does not require accessed or retrieved dates except under rare circumstances when the the web page is expected to update regularly. This is implemented in the Zotero style by including the accessed date when there is not publication date or status provided (i.e., when “n.d.” shows for the date).

when there is no publication date or status provided (i.e., when “n.d.” shows for the date).

(The reasoning is that it's not possible to automatically know "when the the web page is expected to update regularly" so that's the best Zotero can offer. The alternative would be to add [or remove] it manually, which you can do if you want. Another option would to be to use a modified APA style that always includes the date, but then only include the date in your library for items where you want it to appear, but then you'd be missing the date for other styles that might always require it.)

I agree that the obsession with some styles for adding access is unnecessary, and that in general the publication (etc.) date of the paper itself can imply when it was accessed, but there are two cases where it is important:
1. When the website is known to be no longer available at the time of publication. (Personally I check that webpages are current as of submission of a paper, and note any exceptions with a date.)
2. When the data is frequently updated, e.g. a database with daily changes, so that the access date really does change the content in a relevant way. Websites like Wikipedia should also be included in this category, probably, at least if you're citing details.
So I personally don't keep dates except for those cases, although I realize that doesn't match with requirements of all styles. In the case of an editor requiring access dates, I would simply add them back in for all relevant references as the submission date, after confirming the webpages are still available.
(In short, I use the alternative I mentioned above where I simply don't store dates in my Zotero entries unless I know they're exceptions that should have a date. I figure it's easier to add a generic date [e.g. of submission] back in if required than to remove all of the dates that aren't necessary. Not that this would be specific to APA. I'd need a variant of the style if I were to use that method.)

It depends on the type of page: if it's something like a Google search (automatically generated) and you just want to describe it's information, that would make sense. But if it's an official database of some sort with a publication year, but the content is automatically updated, then it should still be dated. An example of this is using a corpus (body of text) in Linguistics, where often there is a publication year (or range), even if there is also continuously (or periodically) updated data. To give a popular example, see the corpora listed here, especially the NOW corpus (updated monthly): https://www.english-corpora.org/