I think the lost nuance is that the previous guy means that with tips, the public pays for tips directly.
You’re technically correct, the public, by buying food and service, is paying the company, creating a pool of money from which the costs of business are to be paid, ideally including staff in full. And currently, wait staff has to be paid by the customer directly.
(This mostly holds for most of the US, in many places, it does work according to the more ideal model)