jacobb NGL you are fighting an uphill battle,
I get this isn't exactly a solution, more of a replacement, but I feel like GOS (or any android OS) won't really solve this issue of yours. If Google Play Store is installed, what stops them from just downloading another browser?
I would recommend if you can't trust your child with the internet, don't give them a communications device. If anything, find a dumbphone that can only call and text.
Avoid phones with communications but parental controls, they have insane security flaws most of the time. No reason to risk your childs privacy like that. The less features the better. I seen phones with just preset dial buttons and not even a screen.
Again, get this isn't really an answer, but it also feels like any answer I give has inherent flaws. GOS doesn't have parental controls features and most parental control features I don't really think are ethical. I get wanting to have your child be able to dial you or 911 at a moments notice. A dumbphone that has 3 buttons, one for mom, dad, and 911 is your best bet, or similar, of course.