Lightning payments are private indeed, although the underlying bitcoin transactions are most likely not (although you can perform measures to anonymize your bitcoin outputs with mixing or other means). So after you’ve opened a lightning channel, and you make a payment, then any information about who receives the payment is anonymous.
Some information is leaked to your channel peers, but using routes with multiple hops over nodes obscure this information, and your peer will not know if you are simply routing for someone else, or if you are making a payment yourself.
When you close your channel you will also reveal that the amounts have changed, but you could have processed a very high number of payments on the lightning network during the life of the channel, so the information revealed is minimal.
True, if you use a custodial wallet, then you are at the wallet provider’s mercy for your privacy, but you can run your own lightning wallet.
So I would definitely say a Lightning Network on Grin would be even more private, but Lightning Network on Bitcoin is quite private as well.