I read an article quite some time ago suggesting the Navy deploy its own land based tankers in areas like the Med and the Persian gulf. It would be much more efficient and leave deck space for more combat AC or allow AC like the S-3 and A-6 to do their primary job instead of hauling fuel. I'm sure interservice bickering hit the roof. IIRC the article suggested using surplus Convair 880s.
The navy primarily uses its fighter class tankers to fill up around the ship. Usually a few thousand pounds. Its organic tanking for additional landing attempts or to keep people aloft in case of a fowled deck, As opposed to the USAFs big wing tankers which are used to extend long range endurance. I doubt there was much turf war because the USAF and other nations share their big wing tankers anyway, and I don't see the USN wanting to spend billions on giant KC aircraft. The navy is a ship force with airplanes, not the other way around. The marines assign land based KC-130s to support their MEUs and since harriers land once everytime, and can land around a fouled deck, L class ships don't worry about organic tanking around the ship, and the herks are used for range extension.
There is always going to be interservice friction, but I think a lot of people would be surprised, especially in an era of shrinking budgets how many capabilities other services are willing to defer, especially if they still get the benefits. The A-10 saga is an example of this and why the army had no interest in getting the A-10. They get all the benefits and the air force pays the bill. Big wing tankers are the same. The USAF is paying the big bucks, but everyone benefits. Fuel is fuel regardless of who owns the aircraft.
Having said all that what seems to make the Rhino different from the S-3 and A-6 is that both of those aircraft were due to be retired and doing the tankING later in their service life. Rhinos have been doing organic tanking from the start, and the fatigue is more of a issue because they have worn out and it's basically a case of "premature aging" and we cant just junk these because that creates a firepower gap