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've never seen that behaviour, it downloads updates in the background and then gives me a little notification that I should restart it to update, but it's dismiss-able.

I've been using Firefox for years and don't think I've ever seen this. Are you running some custom configuration?

Iirc, this is a pattern of the flatpack version of Firefox. I do not believe the .deb does this.

Are you sure this isn't your organization's update policy kicking in here?