Start a no server demo here and you can try Actual out yourself:
https://demo.actualbudget.org/

That way you can try out the Starting fresh steps for real or go through their Tour interactively instead of just reading:
https://actualbudget.org/docs/getting-started/starting-fresh/
https://actualbudget.org/docs/tour/

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.