On your setup.

Firefox does not only not prompt me for updates (it only updates if I manually run an update), even if I do update with Firefox running, it does not stop me from continuing to use Firefox without restarting it. All I did was install Firefox from flathub (packaged by Mozilla (a default configuration provided by Mozilla)).

I am running Firefox Developer Edition on macOS, where it is the default option. Have to set "Check for updates, but choose when to install" to get the similar behavior.