Sitting in a talk at CMU by Bill Tait on cut elimnation for predicative systems. His approach, in contrast to Rathjen and Takeuti, is to try to get the cut-elimination proof to be mostly (or even, only) about the proofs, and not about proofs and (mostly) ordinal notation systems. He’s using the original Tait calculus, in which formulas are all propositional, but infinitary. His cut-elimination theorem applies in all kinds of cases (essentially up through predicative arithmetic), which I hadn’t realized before. But then I don’t think the original paper actually emphasizes that. The main point in that regard is that when you translate axioms from a theory, you can assign the right kinds of ordinals.
Rick Statman made an interesting point, viz., that it’s also important to verify that the infinitary derivations that you get when you translate derivations into Tait calculus are (primitive) recursively described, and that this is preserved when you eliminate the cuts–if this weren’t the case, you wouldn’t be able to carry out the cut-elimination proof in extensions of PRA.