I think this is where you lose me - why is allowing a child access to have un-tethered access to a content portal appropriate? i.e. why is the responsibility on the content portal (YouTube) and not the parent or supervisor?
I might be a tad too much on the individual responsibility side of the spectrum here, but just like the old argument from the 90's ("The TV ain't a replacement for a babysitter...") I'm not sure the responsibility necessarily needs to be on the platform here over the parent/supervisor.
One suggestion: You can create a white-list or playlist of pre-approved content creators and/or videos, and offer access that way.