I mean, aren't diplomats immune to US laws? Literal "diplomatic immunity".

It makes sense to me to say that they do not fall under US laws and US jurisdiction, and their children likewise.

Diplomats have some immunities but are not categorically outside the jurisdiction of US laws. For example, they can be sued for “commercial activity” outside the scope of their duties. If a diplomat sells you fake Hermes bags passing them off as real, you can sue them under civil law.