Oh boy, don't I look foolish now. I'd never even noticed those options before: I'd looked for a setting in table properties instead. It seemed like a problem that would crop up a lot and couldn't understand why no amount of googling various keywords brought up anything like it. Lots of people having trouble with the header row not repeating, but...

And look at that: previously when I wanted banded rows I'd made a table style as it was the only way I could find to achieve it: good practice, perhaps, but for the wrong reason!

Thanks so much!