For example, Firefox updates by default and forces you to restart it, disabling using websites or seeing any site content before doing so.
Not on my machine
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?
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)).