For Chrome, the behavior for me on Windows is exactly as you describe for Mac and Linux. The update happens in the background and button color is used to notify me that a restart is needed to move to the new version.
Firefox on Windows gives me a pop-up indicating a new version is available for download and providing a button to do that now. A restart is still required to get the new version, but AFAIK there's no default visible prompt to indicate the need to restart. If I go to Help -> About Firefox, that tells me I should restart to update...but that's not terribly obvious.
I turn on Linux and Windows machines after 3-6 months of inactivity somewhat often and never hit the issues OP describes, so I feel they are either misinterpreting what they / their relatives are seeing or they have performed other changes to these systems that block the standard behaviors.
Firefox on Windows gives me a pop-up indicating a new version is available for download and providing a button to do that now. A restart is still required to get the new version, but AFAIK there's no default visible prompt to indicate the need to restart. If I go to Help -> About Firefox, that tells me I should restart to update...but that's not terribly obvious.
I turn on Linux and Windows machines after 3-6 months of inactivity somewhat often and never hit the issues OP describes, so I feel they are either misinterpreting what they / their relatives are seeing or they have performed other changes to these systems that block the standard behaviors.