Upvote:10
I'm answering this myself to summarize the discussion in comments, and because no-one else seems to have offered a clear answer.
Original question: Can the child enter the US on a British passport (when travelling with both parents)?
Yes, the child can enter the USA. There are two scenarios for this happening:
Under scenario 2 it's unclear whether the UK passport would be stamped, or if there are any consequences for the parents if they had (knowingly or unknowingly) made a false statement on the ESTA application.
Implicit question: Is this a good idea?
Probably not. Particularly in the long term, several commenters noted that US citizenship carries tax obligations, and by not recognizing one's citizenship, one could fall afoul of the tax obligations and unwittingly violate some law. The parents ought to get a US passport for their child