His Passion, death, and Resurrection all fulfilled many, many prophecies from the Old Testament. The Crucifixion wasn’t a random event. Jesus was Incarnated in that particular time and place because it was the perfect moment to fulfill Scripture and then efficiently carry the Good News all over the known world. He had to be sacrificed, willingly, as a voluntary, sinless Lamb. He had to be raised on a tree so His accusers could behold Him. “By His stripes we are healed,” refers to His flogging and beating by the Romans. If you google about the prophecies, you can find many more.
I can’t speculate on whether a different type of death would have redeemed mankind, but I do know that Jesus was crucified for very particular reasons. Just being hit by a carriage would not have been a sacrifice and would never have happened. God was in control.