The concern with on chain "zaps" is less that the transaction itself is public, and more that the utxo's history and future become public for all to see along with what essentially amounts to kyc at that point in time. Lightning zaps, on the other hand, are public -- however, thanks to onion routing and the lack of a shared public ledger, that is ALL that is public.

This seems to be the krux of the concern. On chain zaps CAN be handled safely, but it adds the requirement that both parties address this with things like coinjoins, swaps, etc, which adds unnecessary costs and of course, room for things to go wrong.