I use rubber bands for competition speed events. As BarrelRacerLvr put it, the bands are "insurance" for not losing your stirrup.
I am a good rider. I have good balance, good seat, and good legs. I very rarely lose a stirrup. However, when you are running Mach 20 at a barrel or a pole and your horse takes a wonky step, I do not want to sacrifice tipping a barrel or a pole because that rare occasion happened during a competition run.
I only use the very thin and very large rubber bands so that I know they will break if I get into a wreck. I know they will because a time or two, I've forgotten to take them off before getting off my horse. Snapped immediately. And sometimes they'll even break just me trying to get them on.
I have enough things to worry about during a competition run. I don't want to add losing a stirrup to that list. So I use rubber bands.
There are some people who use rubber bands as a crutch because they CAN'T keep their stirrups on their own. Those people have no business
using rubber bands. They should never be used to make up for the holes in the rider's training.
Now as far as kids ..... will I let my kids wear rubber bands (when I finally have them someday)? Probably not. I might not even let them have a saddle sometimes.
I want them to learn to ride a horse with balance, and with their seat and legs. Maybe if they are running for a high school rodeo championship, then maybe they can have rubber bands. But unless it is a high stakes run, I'm going to make them go without so they also will rarely lose a stirrup when they are older when they can make their own decision about rubber bands.
I probably didn't start wearing rubber bands until I was 18 or 19, and I feel that I'm a good rider because I learned to be without them first.