For one, your kid's bike likely has a coaster brake. Check that. If it doesn't, either add a second brake or buy a bike with a coaster brake.
Apart from the danger of going over the bars, there is also this: Every vehicle must have two independent brake systems, otherwise it's simply not fit for the road. This is independent of the type of vehicle, and bikes are no exception.
This is both a legal thing (at least in my country, and very likely in any other sane jurisdiction), and a safety issue: Brakes can, and do fail. You don't want your child sitting on its bike going downhill when the cable actuating the brake fails, and not have a backup available. You must have a backup to avoid the worst. I have completely lost track of how many braking cables I've managed to tear, and I've been very thankful for my coaster brake each time.
So, as I said, either your child's bike has a coaster brake, or you need to do something about it.
Daniel R Hicks tells me that some US states allow coaster-brake-only bikes. So, I guess, I have to exclude the US legal system from the list of sane jurisdictions. Maybe they see it as a personal freedom to be allowed to endanger yourself?
Anyway, while your state may allow you to use a coaster-brake-only bike, the safety issue remains: Chains can and do jump off the sprocket, or even break. Especially when they are older and lengthened from wear. If that happens going downhill on a coaster-brake-only bike, you are screwed. If you care about safety, one-brake bikes are just a no-no.