I recall hearing anecdotally . . .
The Shah was sick of the Russkies overflying his country. He asked a Soviet leader if they were Russian planes. The Commie said "no." So the Shah replied, "Then you won't care if I shoot them down, will you?"
Yes, the Phoenix was the key. Computer tracked something like 23 independent targets and chose the optimum 6 to fire upon. Problem with the F-14 on carriers was that if you put 6 Phoenix missiles on the jet, it was too heavy to land. At least that is what I recall 30 years ago.
Also, the AWG-9/AIM-54 Phoenix combination was very attractive to the Iranians. The Soviets had begun overflights of Iran with the MiG-25 and the Phoenix was the one missile at the time which could guarantee an intercept.
That said ... The F-14 was designed around the TF30 but later variants did use the F110 engine which IIRC is physically bigger than the F100. The F110 was also designed to be easily interchangable with the F100. Changing an F-16 using the F100 to a F110 can be carried out by the mechs on base and the only major new part required is a new intake (the F110 requires a larger inlet).
