One of the main issue is that it requires users to create long term links which doesn't make sense if we consider that bitcoin transactions are supposed to be "pseudonymous".
I don't get it. Do you compare the addresses with the lightning invoices? Yes, addresses are permanent and can be used as long as you have their private key, but you shouldn't reuse them. In the LN, you generate the invoice and give it to the buyer. What don't you like about this?
It's far more private than a ledger of transactions everyone can look at.