No, they are selling ad space, which means they're no more "footing the bill for the entire service" than your local CBS TV station foots the entire bill for what they broadcast.
Well, yeah. The TV station also foots the bill.
Just because Twitter has a source of revenue to fund their operations, doesn’t mean they aren’t paying the bills. They absolutely are. The point of me saying Twitter is footing the bill is to point out that Twitter is the owner and operator, through their own expense, using their own money.