Github is one of the few webapps where I can feel daily that the framework used isn't enough. So many things get out of sync/not up to date, which are fixed by refreshing the page.

Very true, but that's a frontend problem, not a rails problem.

These problems are always more common in server-rendered apps though, because front-end state is always a patchwork. And the Rails developers and community have a strong preference for this architecture.

I observe the exact opposite. Server rendered UIs are far more often times up to date than client rendered apps.

I can't think of a single website that's real time and server rendered

I do not have examples with me. But anything built with Elixir Liveview will be server rendered and will also be real time, with some gotchas though.

https://github.com/phoenixframework/phoenix_live_view