I routinely see that users have several old/undismissed notifications in our office when I walk around. I assume this is in part because of similar things to what @tobiaseigen and others have described.
I don’t expect many users to complain about it though. Rather, my concern is that they get so used to seeing a notification bubble permanently positioned over their avatar that they stop noticing it at all.
I don’t think this would completely solve that problem, but it seems to be a somewhat common pattern in apps with similar notification features and it might increase the odds of the feature being used a bit.
So I’d welcome this change if it ever makes it to the top of anyone’s list.