>If helicopters were used since the start, would they have had a major impact on the war?

Production at larger scales impacts production of other aircraft, as materiel and, especially, production capability is not infinite. There definitely is a use case for heliborne assault, but that needs models capable of substantial troop-carrying capacity, and for those to be accompanied by very focused SEAD in order to not become a bloodbath, and there were rather a lot of AA guns that would have found helicopters a juicy target. Even if the helicopters are physically designed, built, and capable of doing something like replacing a paradrop on somewhere like Crete, for instance, the doctrine isn't there. It's not just about the airframes.

You can't use them as gunships because they don't have guided weaponry and they're not capable of outgunning likely targets. Any helicopter flying in WWII would have come up against a panoply of 20mm, 40mm, and larger, AA guns. AAA would have a field day against a helicopter formation. You would need to develop (again) doctrine and training in order to avoid catastrophe, as well as deploying more forward airbases in order to have them within operational range of enemies. The logistics to keep them running are not inconsiderable.

I think they needed to be more advanced. In the Korean War helicopters were mainly used for search and rescue.

But who knows? The Germans (IIRC) scrapped plans to station gyrocopters (they're an earlier form of helicopter) on their ships on the outbreak of WWII. Maybe if they had used them, other nations would have followed suit and made more advanced helicopters. There was a lot more money for R&D during WWII compared to the Korean War.