I've setup exactly what you've described. Some of the stories of people suddenly losing access to their Google accounts scared me enough that I wanted a copy of all my emails locally in case the same thing happened to me.

I setup an IMAP email server using Dovecot which I can access over my Tailnet from various client machines. I use Getmail on the server to periodically poll Gmail every few minutes for any new messages via POP access. Anything new is downloaded and dumped in the user's inbox.

It's a pretty solid setup - I've been using it for years.

"And carriers still charge 2 euro for a MMS which completely kills iMessage/RCS (if accidentally send MMS, it's expensive"

In signal why not just turn off resend as MMS? Problem solved.