Is there any way to get push notifications instead of polling? My email provider is Fastmail, maybe there are some kind of IMAP extensions that can do this?
Yes! They just moved the settings around compared to K-9. You have to open the folders side bar, hit Manage Folders, then tap your folder and turn on "Enable Push" (and probably Sync and Notifications too). Annoying that you have to do this manually for every folder but it does work great!
How would a true push (and not poll) method work, if all you have is the local thunderbird client? If I had to guess, the question from the other person is not related to mail poll/push stuff
Usually you rely on a server telling the mobile app that there's something new to poll (or the message is directly sent and displayed).
A TCP connection to the IMAP server is kept open, with the server sending a packet whenever anything interesting happens (see IMAP IDLE), waking up the client app. This is how all push notifications work nowadays, IIRC, since you can't send information directly to clients without a persistent connection, due to NAT.
Fastmail is a webapp, and it's very obvious. I use it out of laziness, because I don't use email on my phone that much. I wish they'd make a proper mobile client, but I don't think they care, since it's been like this forever.
OT1H, I don't have a horse in the fight about whether you're right or not, but I was curious because if it is a web app they did a damn stellar job
But no, it's a formal android app and I offer two pieces of supporting evidence: the resource browser shows the apk very obviously has a bazillion android resource files <https://imgur.com/a/pUbQ8Bp> and it supports native fingerprint auth
True. I seem to remember that the UI itself did not load when there's no internet connection, but I just tried it, and it does now. So maybe things have changed, or I misremembered.
Either way, no offline access in an email app... pretty bad.
It's not a native app. The main interface where you view your mailboxes, compose new emails, edit settings etc is a web view. The interface doesn't even load if you're offline, let alone your emails. Certainly some parts of the UI are native, but that's true of any web app.