Ah, after restarting after the update process the setup utility is not autostarted.
And running System update again now does the update process immediately, no waiting this time.
But what if I was a new, regular user and didn't wait last time? What if I hadn't even noticed the system update tool at all, seeing as it's hidden under a Utilities submenu in the applications list? Would Bazzite prompt me to update after a while, or would it just let me run with no updates forever?