This doesn’t really seem like something that needed to be done. Thunderbird already has too many features. It needs less, not more. A bunch of stuff in the email client part is also badly designed. That needs fixing, preferably upstream, but I wouldn’t think of that as feature enhancement.
I use TB under Debian and there is a tray icon and an arrival notification, poll time of maybe a few minutes, seems fine. Showing the # of messages in the tray icon could be sort of handy I guess, though I had never thought about it before and didn’t miss it. Basic features = shut off the “email contains remote content” banner or “spam filter thinks this email is spam” (I can recognize spam for myself). I just want a preference that permanently disables remote content without throwing banners at me. And eliminate the client side spam filtering completely since I have that on the server side, and can manually flag any that gets through. Plus various other stuff like that. Yes, get rid of the calendar and contacts stuff. Biggest feature needing significant code changes: make message search not suck.