Don't think he has what it takes for the very top level, too much of a weathervane politician. If you're convincingly beaten by Jeremy Corbyn in a popularity contest you probably need to re-consider your Leadership aspirations. Not that we can lay the Corbyn fiasco at Burnham's door, at all, but it is interesting to think that in most universes he beats him handily, with JC putting in a credible showing and re-establishing the left on the back benches. Just not this one.
Burnham does have obvious qualities, though, so will have learnt from that - maybe he can go again.