Selfhosted

99.9% of users don't have/use "server-side sieve filtering", so every new mail comes to the inbox only and the user might move it to a different folder later on. Because the workflow of most users is like this, you will have a hard time "going against the grain" if that makes sense. My practical recommendation to you would be to just use a single inbox like everyone else if it's not critical for you to have "server-side sieve filtering". I know it's hard sometimes to not have something work exactly like you wanted it. It happened to me many times also. Going with the majority is much easier and less time consuming than going the other painful, lone, hard route imo. Anyways, hope you find a good solution :)

SnappyMail seem to be a fork of Rainloop and both Rainloop and Snappymail appear to allow multiple providers - https://snappymail.eu/
Cypht seems to be a similar solution where you selfhost a webserver that acts as a web client to external email providers - https://www.cypht.org/documentation/
I find nothing about push notifications for either of those solutions though, and I'm not sure about how much the webclients cache.

Thunderbird on Android should support push, have you tried that?

If I understand correctly, you want a two component setup. A PWA client for you to read the mail, and a server acts as IMAP client, fetches mails from all you mailboxes. The server will expose an API for tge PWA to access mail content. When new mail arrives, the server push a beacon via the Push API. The PWA would fetch the sender and title, and display a notification. If you clicks it, only then the PWA will fetch the body.

After a quick glance of the demo, I think SnappyMail fit the bill? It seems can be installed as PWA, and my browser does ask me if I want to give it push notification permission. However, I'm not too sure if the fetch logic happens as I laid out.

Haven't tried it yet, but I am going to attempt https://mailcow.email/

Might be easiest to just find a mail host that supports push notifications and keep using the mail client that works for you. Unfortunately, I don't see how you're getting a webmail client with multiple mailboxes without hosting that yourself with something like Snappymail. Maybe someone offers a paid and hosted Snappymail.

I host my own mailcow server and enable notifications for mailboxes I want to get notified for via Pushover. I have Snappymail in the stack, but rarely use it because I like K9 on mobile better.

The point of a mail server is that many clients can connect to it and get what they need. What I'm reading here is that you want to disregard the ability to already do that in favor of having all your mail funneled through a self-hosted 'something' that just sends push notifications to your mobile...but then you'd still have to have your mobile mail client go and download all this mail you said is a battery drain, so you're sort of negating yourself.

Now...the real crux of the problem you're describing is simply that your mobile mail client is not very efficient, so why wouldn't you just solve for that instead? Create a better workflow for your mail so your client doesn't need to IMAP crawl EVERYTHING, or reduce the frequency it syncs maybe.

If that's still not enough, depending on your mail host (which you didn't mention), there are ways to simply subscribe to push notifications from their service more than likely if that's all you want.