Why do citation styles for webpages include accessed date? I do understand that citations should be as precise as possible (including e.g. edition of a book or a page number), so that ideally, the reader can find the information exactly as I found it when writing.

I also understand that the problem with citing webpages is that they can change. But how does including the date when I accessed the page help anything? The reader still won't be able to read the same version I did (unless the page includes history, which is very rare). And comparing the accessed date with the current date isn't very helpful either: some webpages change multiple times per day, some aren't modified for years.

Or is the accessed date useful for some other purpose?
A URL does not refer to content, it is only an address.  Since you are making a claim about what was written at that address, you must associate a date with it, or it is meaningless.

If you do such a thing, you should also save the content, in case it is not recoverable and you want to refer to it again or support it further.