chrysmon The question is what is your priority. If it's about convenience and something closest to Activesync, then look into JMAP. Here's an example client.
As for a more traditional solution, FairMail is the best. K9-Mail/Thunderbird are convenient to use. Ready out of box are Tuta and ProtonMail. Tuta has notifications, but Proton still doesn't seem to have notifications on GOS (EDIT: see this).
If privacy is important to you, then instead of IMAP choose the old POP3 and a service that allows you to encrypt received emails with a public key (if they are not encrypted). This way, the server will never keep messages in an unencrypted form. POP3 because it allows you to completely download messages from the server.
If you expect anonymity, then don't use email or take a look at GuerrillaMail, you may find a client there as well. Use Tor.
It's best to just not use email.