I find them useful in some cases. I play board games online, and both sites I use will update the favicon (either with a new color or an animated shape) to tell me that it's my turn.
Sounds like that use case really wants something like silent push notifications (no sound, no toast, but it's there waiting when you decide to pull down and look for updates from all apps in one place). Yes, I said push and pull to describe a sort of hybrid model. Android offers this, but I'm not sure if desktop OSes do.