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For sure - as I understand it in most formulations you throw out lots of sets but keep all the finite sets of natural numbers, the set of even numbers, the set of prime numbers etc

One thing I never found satisfactory is that any axiom system like this has to define things in terms of decidability etc, so it's "more verbose" (or less axiomlike) than ZFC

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This is a fair point.

I think the only axiom you need to rethink from ZF is powersets (since I think that's the only axiom that produces uncomputable sets from computable ones (ignoring the AC, briefly)). What you'd replace it with (some sort of one based on comprehension, presumably) I couldn't say though.

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