If the Marine aircrew had gotten involved at an early stage, the issues with training would have been resolved. OPSEC concerns got in the way of that, along with only eight helos aboard Nimitz: the AAR mentioned that the plan should have allowed for possible aborts, and thus have 12-16 helos. OPSEC again: there were worries that if the Soviets saw a dozen or more RH-53 on the carrier-which didn't normally embark them, the GRU would start adding things up, and possibly warn the Iranians.
The actual raid on the Embassy would've gone off without much of a problem: the raiders had accurate intelligence as to where all the hostages were being kept (in the chancery), and could ignore any other buildings on the compound. Throw in at least one AC-130 orbiting over the Embassy, ready to give fire support on call (one more was headed to Meherabad AB to kill the F-4s sitting strip alert, with an airborne spare), and Navy tac air from CVW-8 (Nimitz) and CVW-15 (Coral Sea), and the raid might very well have gone off. Casualties were expected, up to fifteen hostages, and it was this possibility that led SECSTATE Vance to resign.
Carter was a bit naïve: he asked Charlie Beckwith if they could use some kind of knock-out gas to incapacitate the guards, and when Beckwith said no, sir, there's no such agent, Carter asked if the guards would just be shot in the leg. Beckwith thought it was a joke, then replied, "No, sir. Anyone armed gets two in the chest and one in the head. No exceptions." One gets the impression from reading both Beckwith's book and Jim Kyle's The Guts to Try (USAF Col. Jim Kyle was the senior USAF officer on the mission, leading a Combat Control Team), that Carter wanted the hostages rescued, but no Iranian casualties.
EAGLE CLAW was like going into Hanoi and busting POWs out of the Hanoi Hilton. (and yes, there was a plan in 1972 to try just that, but it was rejected)
Edit: the only USAF helos that could have flown the mission were HH-53Cs, and they did it before: the Son Tay raid in 1970. The helos were capable of air refueling from HC-130s, and they did just that on the Son Tay mission. IF the AF gets the helo mission, they refuel in the air and no Desert One fiasco. However, you may not get around the need for a hide location, as aircrew fatigue would have become a factor-which leads to mistakes, which leads to accidents and mission compromise.
Egress would have been via Manziyareh, an Iranian AF field used for weapons demos during the Shah's reign, and a Ranger Company was tasked with securing that field. C-141s were to fly in and embark the raiders, freed hostages, and the Rangers, then any remaining helos were to be blown up (thermite bombs are good for that), then the C-141s, with Navy fighter cover, were to fly to Oman.