"But for computational systems... there’s my Principle of Computational Equivalence—which implies that all these systems are in a sense equivalent in the kinds of computations they can do." Am I missing this, or is he restating the Church-Turing thesis but taking credit for it himself?

I hope not. But Wolfram had been accused of this sort of thing before, see e.g. https://arxiv.org/abs/quant-ph/0206089