Only on the surface. It was a powder keg for years. And the Shah was a butcher in his own right. His secret police [Savak]were.
I don't know that we had many options in the middle east at the time because our policy was guided by cold war realities and the encroachment of the USSR in the region.
But don't kid yourself regarding Iranian stability under the Shah. It was an illusion for many years before 1979.
The Shah's problems started in the mid 70's, when he declared himself "emperor" and usurped the authority of the mullahs (or at least made them think he was going to) in religious matters. That's when SAVAK became really oppressive. Throughout the 50's and 60's, there was relative stability.