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.

I definitely feel the pain when it comes to worthless results nowadays. Though in this case DDG comes through:

Adding documentation to the search makes the “correct” page soar to the top: