A democratic Iran will always be the enemy of Israel.
Israel is a nuclear-armed ethnostate engaging in ethnic cleansing that protects these transgressions by operating as the FOB of the hegemonic world power that doesn't blink an eye at killing a million citizens in the region.
So long as Israel requires US support to safeguard their otherwise untenable position (and to force Arab dictatorships to act favorably towards Israel), Iran and Israel will be opposed.
Iran and Israel were allied before the revolution because the undemocratic government (not that the current one is anymore democratic otherwise) that was willing to act against its interests on the matter so long as the dominant foreign powers required it.
It's categorically false that no one gains from the conflict. Israel gains tremendously from the conflict as the antagonism between Iran and Israel and between Iran and the USA is what justifies its ability to get away with so much.