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RichardKennaway comments on Deliberate Grad School - Less Wrong
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Yes, ZFC may be not quite such a starkly isolated landmark of thinginess as computability is, which is why I said "a strong tendency". And anyway, these alternative formalisations of set theory mostly have translations back and forth. Even ZFA (which has sets-within-sets-within-etc infinitely deep) can be modelled in ZFC. It's not a subject I've followed for a long time, but back when I did, Quine's NF was the only significant system of set theory for which this had not been done. I don't know if progress has been made on that since.
(ETA: I found this review of NF from 2011. Its consistency was still open then.)
As for computable functions, yes, the different ways of getting at the class have different properties, but that just makes them different roads leading to the same Rome.
Randall Holmes says he has a proof of the consistency of NF relative to ZFC (and in fact something weaker, I think). He's said this for a while, he's published a few versions of his proof (mostly different in presentation in the interests of clarity, rather than patching bugs), and I think the general feeling is that he probably does have a proof but it hasn't yet been thoroughly checked by others. (Who may be holding off because he's still changing his mind about the best way of writing it down.)