If a child is misbehaving in the lounge such that it substantively negatively impacts my ability to enjoy a lounge, the problem almost always is a parent or some other adult in the lounge.

Most children in hotel lounges have been no worse IMO than the far more annoying adults in the lounges. And most parents seem to handle their children rather appropriately in the lounges.

If hotels were to prohibit cell phones and TVs in hotel lounges, then perhaps I'd find some consistency of position with that of prohibiting young children.