Actually, they do have to pay for some things and the servants, chefs, etc. are partially for the executive mansion itself, other employees who work there, upkeep, dignitary visits, dinners and lots of other things. It’s not just a private family residence, it is also a large office complex, diplomatic headquarters, communications center, military facility, press mélange, museum, government compound and so much more.
I understand that it’s quite an operation, and has to be.
But I never heard that the President must pay personal expenses. Do you have more info on that? Link perhaps?