Yes and pushing 'when do i update to/from the server' onto the user is exactly what I'm complaining about. No other software I use does this. Even software originally built to do things locally allows you to seamlessly update to/from a server nowadays (e.g. working on a cloud document in Word say). And any software built from the ground up to do this tends to have a great user experience (e.g. Figma).
You can absolutely build a system that fetches a git remote whenever it updates, you just need some sort of a notification/subscription system to know when to trigger it.
You can absolutely build a system that pushes (some of?) your branches on every commit.
For me personally, that'd be annoying; every time I talk to the git server, I want to insist on a Yubikey touch.