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I haven’t used Manjaro myself but I heard that it is not as good as Arch. Rumors I heard where that it is not as solid as vanilla Arch. YMMV.

Which one(s)

Arch.

why?

1. The Arch-Wiki
2. I like pacman
3. The Arch-Wiki
4. I wanted a rolling-release distribution.
5. The Arch-Wiki
6. It just works. I had only one more serious problem in ~8 years of running Arch
7. Did I mention the Arch-Wiki?

Edit:

Having said that, I have an eye on immutable distros. Maybe one day I’ll try one out.

It has nothing to do with Trump. Even theoretically that can’t be the case European budgets haven’t suddenly changed in the last couple of days.

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).

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

is there a simple way to get thunderbird to fetch them from the moment I turn the pc on and give me notifications about it?

Sure. You can autostart Thunderbird and keep it open but I haven’t found a way, where Thunderbird closes/starts to the tray and for some odd reasons the developers seem to think that users do not need this functionality which makes the whole email client unusable for a large part of the potential user base.

I can get the clients to fetch the e-mail atm, the issue is what I wrote above,

??? You wrote:

Who the heck knows how to get evolution/geary to play nice with business gmail/protonmail.

gmail

I don’t know much about Gmail but I’m quite certain, that you only have to enable IMAP/SMTP in Gmail settings.

protonmail.

Install the Proton Mail Bridge and connect to the IMAP/SMTP server on localhost (ports 1143 and 1025).

Does anyone have a simple way of solving this problem?

I had only minor problems getting the above to work. Anyway., for Protonmail there is ElectronMail. It’s available as Flatpak too and it minimizes/starts to tray.