Well, since it's football season here in North America, I decided to do a good ol' end run.
I realized that, in the final analysis, what I really wanted was simply for all my email, from all my accounts, to work well with the Apple clients on macOS and iOS; and then I further realized that a lot of my aggravation with email and those clients would go away if I simply bit the bullet and made Gmail cease to be the elephant in the room, since its non-standard ways and anti-Apple practices were causing most of my problems in this regard.
So, after some research into how others had done it, I got my own domain plus an account with FastMail to host it for email, and then migrated all 12 years' worth of Gmail content (and set Gmail to forward any subsequent emails) to the new place.
Of course, this had its own upfront pain to handle. I had to notify quite a few vendors, friends, et al.; I'll have to rebuild a lot of rules for automatically moving some emails to folders (real IMAP folders, not the Gmail bastardizations thereof); and I'm still also keeping the Gmail account live partly because I want to make sure I haven't forgotten anybody and partly so Google won't shut it down, because that does happen and then you can't ever get the old account name back.
...
BUT FastMail comes highly recommended, costs only about 14 cents a day (plus about 4 cents a day for the domain, at least where I bought the domain), and is far more standards-compliant than Gmail, which means it works a lot better with Apple's clients including, yep, providing absolutely normal push. Stuff comes through almost instantaneously. And, best of all, if I ever decide that the Apple clients
aren't what I want to use, the standards-compliance of my new setup should make it highly portable.
I will be the first to admit that such a move ain't for everyone, but that's my current version of the "holy grail": rather than gripe because the car bounces on the hole-filled road, I decided to drive it on a smoother road.

And, so far, I'm enjoying the ride quite a bit.