Seems to me it’s easier to just spend more for an extra Nimitz.
In rational terms that's pretty much it, and is what the USN actually did amongst other things like retain the Coral Sea.
However the 'just' is not a reflection of that path to get to that 4th Nimitz. The USN had to go through a prolonged process to kill the CVV and the SCS between 1970 and 1976, and perhaps somewhere within this political turmoil the acquisition for the Eagle to cover a short term (10 years?) requirement might be a solution.
If the Navy needed a carrier that badly, they would have just modernised Oriskany and maybe Hancock while also refitting FDR. Too much on Eagle would have been a "one off" item making maintenance a god awful expensive nightmare.
Again the 'just' doesn't do justice to the issues involved with the Oriskany and/or Hancock. The obsolescence and dwindling size of the F8 fleet is something we've discussed at great length in the Essex thread, as well as the seaworthiness issues that the Essex and Midways were experiencing as a result of their extensive modifications. The Eagle might well do an end-run around these issues for a short (in carrier life span terms) time.
More to the point, Eagle only carried 34 fighter/strike aircraft to an Essex’s 60. Albeit 14 of the British strike aircraft were Intruder-sized Buccaneers instead of Corsairs or Skyhawks.
60 aircraft was in the early 60s, by 1972 an Essex's CVW was 48 F8 and A7. I imagine the USS Eagle's CVW would be a sqn of F4 and 2 sqn of A7, the F4's higher performance and all-weather capability making up for the lack of numbers. As for the F4's performance in the Eagle, I think the F4S version would address some of the issues that the F4B had that caused the British to develop the Spey Phantom; it certainly had more powerful engines and aerodynamic improvements to the wings.
Do please note that the British used many parts from HMS Eagle to Keep her sister ship HMS Ark Royal in commission till 1979 so by doing as the OP suggests effectively mothballs Ark Royal far earlier.
True, but that is in large part making virtue of necessity; the Ark had only been given a 5 year reprieve so cannibalising Eagle is both needed and useful. If the USN picked up Eagle and was going to sustain her for 10 or so years the sustainment cost would be divided between the USN and RN to manufacture the spares rather than strip Eagle, and when Ark pays off the USN would strip her to get some spares for Eagle.