As simple as the title sounds I'm having huge trouble getting that working.
Thunderbird only fetches new mail while it's open.
Who the heck knows how to get evolution/geary to play nice with business gmail/protonmail.
Does anyone have a simple way of solving this problem?
edit. Also, somewhat related, is there a good looking, simple e-mail client? Thunderbird looks busy. Geary kinda looks okay but I cant get it to work at all.
That's weird. I run Geary myself for a couple off accounts and so far it does the job perfectly and without hiccups.
Anyway. You may try birdtray as written in one of the other comments but I'm pretty sure I tried it at least once and for some reasons wasn't convinced. YMMW
Do you get background notifications with Geary?
Yes. If it helps: I run it under Gnome. Maybe you need some extra service running?! I just checked and on my machine - in addition to Geary - there is the evolution-data-server running among others (evolution-source-registry, evolution-alarm-notify, evolution-calendar-factory, evolution-addressbook-factory).