Friedman showed in a 1971 paper that any proof of Borel determinacy requires the axiom scheme of replacement.

@A.C. try Takeuti's book Proof Theory (second edition). He refers to $\mathsf{ACA}$ as $\mathbf{S}^2$. The system Carl has talked about ($\mathsf{ACA}_0 + \Sigma^1_1\text{-}\mathsf{IND}$) is referred to as $\mathbf{S}^1$.

A quick look makes me think the thing that's wrong is your contention that "if you consider predicative second-order arithmetic with the ramified hierarchy going to arbitrarily high transfinite ordinals, then the sets you'll ultimately get are the hyperarithmetic sets". Kleene proved that if you iterate the ramified hierarchy up to $\omega_1^{CK}$ you get the hyperarithmetic sets. But that doesn't exhaust the sets of natural numbers which one can get from the ramified hierarchy by iterating through more ordinals.

There's a nice survey in Feferman's chapter on predicativity for the Oxford Handbook of Logic and Philosophy of Mathematics. You can find it online here: math.stanford.edu/~feferman/papers/predicativity.pdf

@DavidSpeyer you're describing $\Sigma_{n+2}$, not $\Sigma_{n+1}$.