I agree that there's probably a better way, but I fear that it could be at the cost of either ease of use or methodological clarity. Ranking candidates is relatively intuitive, and so is sequential elimination with votes reassigned to the next preference.

I recognize that you could transform the rankings people provide into head to head match up, but if the methodology is too opaque to the layperson, it's going to breed distrust in the system, and I fear that that would be overall worse (if it discourages participation) than having a minority of elections be won by the 'wrong' candidate versus better approaches.

I recognize that this could also be an argument in favor of FPTP. I do think there's a continuum here of conceptual difficulty, and I think finding the sweet spot of correctness vs intuitiveness is tricky.