It would be a much bigger change than moving to alternative voting or majority rule.
Alternative voting, in this example, doesn't work very well, because it eliminates the majority winner, but by and large it's a much better system for capturing the majority will than first past the post. I think either alternative voting or majority rule, or some other similar variant in which voters have the opportunity to express themselves by ranking rather than just voting for a single candidate, would be a considerable improvement over the current first-past-the-post system.