geary/evolution seem to just… break sometimes
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
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).