The lag in vaccination appointments is driven by constrained supplies. Nobody will be given the option of paying to jump the queue although it wouldn’t be a surprise if the PL have thought about this idea. In their heads everything is for sale at the right price.
As to whether club doctors could administer, the answer is yes. Pharmacists in Boots can administer vaccines so no reason why a doctor couldn’t do the same but they should be vaccinating the general population not their employees.
Not so sure. It is about getting the vaccine in the hands of a doctor or nurse (not the rate limiting step I don't imagine), then getting the punters to form an orderly queue for the jab to be administered (the rate limiting step I suspect). I don't have a problem with the club doctors jabbing the players. It would be even more lovely if, once they are finished, they could spend some time jabbing members of the public too.
Anyone with private health insurance been informed how long their queue will be?