I think Iran's posture toward Israel has much to do with the antagonistic relationship between Iran and the U.S. The U.S. allied with Egypt and Saudi Arabia long before those countries recognized Israel's so-called right to exist, and even during times when those countries actively supported militant groups hostile to Israel.
Egypt's and Saudi Arabia's stance toward Israel has softened partly because of their relationship to the U.S., and partly because of the bifurcation of regional interests--the enemy of my enemy is my friend.
Turkey, notably, has recently vacillated between strong support of Israel (including bilateral--U.S. not involved--strategic security cooperation) to being passive-aggressive (i.e. ignoring arms shipments). Not coincidentally, Turkey has had a similar relationship with Iran. And through all of this Turkey has remained a NATO member.
So... it's complex. The U.S. is perfectly capable of normalizing relationships with a country hostile to Israel, as long as that hostility isn't openly flouted. The degree to which Iran openly flouts its hostility to Israel, and to which the U.S. stylizes itself as Israel's protector, is context dependent and self-serving and really not about Israel, per se.
All of which is to say, the U.S. is antagonistic to Iran not because of Israel, but because of the U.S. And vice-versa. And it all quite clearly started with the Iranian Revolution.