I don't think they're this dumb, just bought by Israel. Same with the Iraq war that was obviously bogus.
As for the Iraq war, Israel was supportive, but it would be incorrect to say the war was at their behest.
Well also, about Pahlavi being deposed by his own people, who really knows.
They also need to maintain Western support. That's billions in both military aid and payoffs to neutral countries like Egypt and Saudi Arabia, plus diplomatic ties. Netanyahu's administration wants to maintain support from its own people too. Having enemies isn't a completely bad thing for those reasons.