Proponents of the TI (and the PTI) explicitly say that one advantage of their interpretation over the MWI is that it is a one-world theory

Even the PTI? With the given description, that's really really weird.

Simply saying, "Well, in MWI observers will see all sorts of different frequencies in experimental results, including Born-rule compliant ones" doesn't tell us why observers should expect to see Born-rule compliant frequencies in a QM-governed world.

Not ALL sorts of frequencies. Many - most - ways of looking at the wavefunction won't reveal causal structures isomorphic to observers. For instance, you can consider the wavefunction in the energy basis and interpret it as an infinite number of wheels of various sizes, spinning at different constant speeds. No observers are apparent when viewed this way.

I suspect that the Born Rule is the only rule that leads to observers, but we don't need to prove that it's the only one, and I'm open to the possibility that there are others. GAZP again - if it's in there, it's in there, whether or not you're aware of it.

And if you don't have a story about that, then I don't see how you can claim that the observed frequencies confirm QM (and by extension MWI).

I think you'll agree that the observed frequencies confirm the conjunction of Schrodinger's Equation with the Born Rule. The question at hand is whether the Born Rule needs to be a rule of the universe. Whether collapse is ontologically real or based solely on our parochial viewpoint as observers.

Suppose it is real. That's nice. We get everything we see.

Suppose it isn't, and collapse isn't a real thing. The wavefunction is just doing its thing, and that's all there is. The causal structures in the wavefunction that correspond to people are still there.

The way of looking at us that brings us into focus is the Born Rule. Removing the Born Rule is just like removing a P-zombie's consciousness. It's that switch you flip to grant or remove subjective experience from a computation that implements consciousness.