Would a system that did something like an internal cut-over be useful? e.g. try to start a whole new instance of the application, if it loads, then let it become the running application, if not, write an error log and shutdown?

It would still lose all the state associated with the previous instance, e.g. user sessions, but would avoid this specific issue.

I agree that it's pretty silly that things like email addresses need a restart, but I'm wondering in general how bad this pattern is.