Hold on a moment. What ladder is being kicked away here?

We've got observed probabilities. They're the experimental results, the basis of the theory. The theory then explains this in terms of indexical ignorance (thanks, RobbBB). I don't see a kicked ladder. Not every observed phenomenon needs a special law of nature to make it so.

Instead of specially postulating the Born Probabilities, elevating them to the status of a law of nature, we use it to label parts of the universe in much the same way as we notice, say, hydrogen or iron atoms - 'oh, look, there's that thing again'. In this case, it's the way that sometimes, components of the wavefunction propagate such that different segments won't be interfering with each other coherently (or in any sane basis, at all).

Also, about density matrices - what's the problem? We're still allowed to not know things and have subjective probabilities, even in MWI. Nothing in it suggests otherwise.