Consider how some developers today talk about "toast" notifications: they understand on some level that calling them pop-ups is bad, but came to a consensus that using a different name was appropriate, rather than not actually making pop-ups in the first place.

Theme Hospital (a funny 90's game) had a cool way to do notifications, watch on the bottom left as icons pop up: https://youtu.be/26O35BOTVSI?t=693

Some disappear if left unclicked, I guess if the problem is no longer relevant (e.g. a patient bored of waiting leaves), and they can be attended to in any order.