Interesting: On MacOS, with the option "Check for updates but let you choose to install them" chosen in the Firefox preferences, the experience is exactly that, update in the background without further notifications. That is, I get a prompt that an update is available and whether I want to download it now or not. If so, it starts a download in the background and applies the update on the next start of the program. (You can also check for updates and restart in the "About" dialog.)
(Note: I've set new tabs and windows to open with a blank view, which may also make a difference, as there is no default content.)