But the recipient still is publicly marked as having received BTC, right? It's just not necessary to zap for allowance and many other types of payment.

And a completely private zap would maybe be good as a payment receipt but then it wouldn't be a zap, would it? 

 if you don’t want to publicly receive zaps you would just disable allowsNostr on your lnurl endpoint. This is a hint that clients use to make sure it sends regular lightning payments instead of a zap.