I actually don't think ZKPs solve the age verification problem (for a variety of reasons that won't fit here), but they solve plenty of more interesting problems. Private transactions in Zcash, for example.

Only because zcash trusts the sending node - whoever is hosting that ledger.

If they had no clue who it was, like the typical http web, they could not allow a sender to be a prover, since they would not be able to verify their balance. At some point a balance inquiry would have to reveal who they are. It only works if you trust the banker.

Or by being a middleman. You send [payment service] the money and they send it to the recipient, hiding your info.