You have to understand that the animating spirit of any majoritarian system is essentially like that of the existing system. It just does it better and with fewer anomalies and, in particular, is less likely to produce what an academic would call “social choice perversities”. It's not perfect. It's always possible that the true second choice of a majority of people on the ballot is eliminated before you can get to that candidate, but I think that's a relatively small problem.
If the majoritarian framework is the preference of the broad range of people, then I think the alternative vote is the way to go, because it does ensure that, as fairly as can reasonably be said, the person who emerges as the winner is preferred to the alternatives in a straight fight. That's, in some sense, the essence of functioning majority rule. It doesn't produce proportionality, make no mistake about that. It could produce some amount of functional coalition behaviour, but the record suggests not really.
The recent record in Australia, where governments of the right in particular have failed to get majorities, partly reflects other features in the system, in particular the existence of the more or less proportionally elected senate, which has had the effect of dissolving some of the claims of the big parties in House elections in Australia.
But in terms of general fairness, I do think that if you're operating within a majoritarian framework, a majority ballot is a better ballot than first past the post. I am of that view.