Yes you pay LN invoice from cash balance. The service strike offers there is whatever they have do behind the scenes to “be compliant” and make the payment happen without putting any kind of tax burden on the customer, etc.

Why have a buy button, well that’s a good question. One could argue it has a similar answer. More likely though is that people just like buy buttons and the invoice trick is a little bonus for those who understand and are willing to do a couple more clicks