It's really a you problem. The market share of Linux desktops is low enough that in reality it doesn't matter, and you're pulling the rug(binaries) from underneath a web browser (one of the most complex software projects known to man) and expect it to keep running flawlessly.
I'm using NixOS where every version of every dependency of every program I install will be saved until I cleanup the garbage (1 command) so I don't have your issue.
If this was Windows you wouldn't be able to update Firefox while it was running because Windows doesn't support overwriting open files (technically you can rename them and insert new ones with the same name, but that's not the point).
You're holding it wrong and blaming the project, close the software you update before you update and you might have more success.