I'm kind of curious but when is the last time you ever had any sort of sustained interaction with a child?Why don't we call kids who are younger than 8 by the last 2 digits of their social security number in situations where there are many children (classroom setting, etc.). Family vocabulary can be used at home like "son", "daughter" etc. if this is too clinical sounding for parents.
To answer your question about foisting the responsibility of choosing a name on a child. I don't imagine this question would be suddenly thrust upon a kid without prior preparation by parents/ teachers/etc. I was raised Catholic and received first communion around that age. I had to take CCD (religious education) classes leading up to receiving the sacrament for the first time to understand the importance of it as part of the religious ceremony.
Likewise, choosing a name would be treated like an important societal rite of passage. Parents and teachers would confer the importance of the event on the child and offer guidance well in advance of the choice. Additionally, as I stated in the OP, the child would be presented and taught hundreds of possible names in the months leading up to the ceremony to make it less likely they name themselves "Roblox" or whatever they're interested in at that moment (however if a child was set on being called Roblox they could do that).