If you value your player, you'll choose the best. The whole point of this facility will be for it to become a centre of research & excellence.
When I badly damaged my knee playing football in 1988, my Consultant was a Dr Hurst. It was during a conversation he revealed he was also the main surgeon used by both City & ManUre. I assume both clubs thought him the best available, & again I assume neither had an issue with him being consultant to the other.
If I was badly injured, & a Dr returned my bitter enemy to full fitness, why wouldn't I choose him to repair me? Unless you're suggesting a 'City affiliated surgeon' would nobble an opponent player for City's gain. If not, I can't quite see the point you're making mate.
Also, I doubt City will be looking on this facility to significantly add to our balance sheet. The facility is what it is.... A place of medical education, learning, research & excellence, right on our doorstep, which adds to the aura of Sportcity.