I suspect performance and inconsistent experience killed Java applets and Sun stopped caring about desktop Java when they realized they can't make much money off it. Flash stuck around a lot longer than Java because it was fast, fun and ran consistently. It still lingered for some time after Jobs poisoned it.
Honestly, I don't think people cared that much about running updates. For most people, if the computer tells them they need to update, they update. I also don't think average users cared about security issues back then.