I saw this post by Carl Mummert, which describes how one can 'see' the reasonableness of ZFC, via iterating the power-set operation starting from the empty set. However, this iterations proceed in stages where each stage is from some well-founded collection. We can intuitively grasp finitely many iterations, and given our intuitive assumption of natural numbers we can also grasp their union, and we can repeat this through all the computable ordinals, but I can't see how one can justify that anything beyond that is reasonable besides simply appealing to the ordinals already existing, which would be circular if we base our understanding of ordinals on ZFC.
Specifically, to reach the first uncomputable ordinal we have to have all the computable ordinals, but these are in a sense not all there in the above 'construction', because if our goal is to justify the cumulative hierarchy we can't talk about any uncomputable ordinal since we have not constructed them yet, nor can we make use of the collection of all computable transitive well-orderings because it is merely a concept. Sorry if this is quite vague, but I don't at the moment know of a clearer way to express this meta-theoretic concept.
Also, if ZFC has a (countable) transitive model, then there already is a countable ordinal that is a model of ZFC, whose existence therefore ZFC alone cannot prove. But I think this is far beyond the first uncomputable ordinal, though it seems to me that it prevents non-circular justification of the cumulative hierarchy beyond it, since we don't even 'know' whether it exists without assuming more than ZFC.
So my main question is:
Is there some meta-system that isn't stronger than ZFC (but might be incomparable) but can construct the cumulative hierarchy and hence provide some kind of 'justification' for ZFC?
If not, what is the furthest we can go (or know how to) up the hierarchy?
I apologize that I don't even know precisely what meta-system can allow my description of the computable ordinals to go through.
Finally, the post by François G. Dorais that says that we cannot obtain the first uncountable ordinal without the use of the power-set axiom, which would present a severe obstacle if we do not have the power-set operation (for infinite sets). To me the full power-set is also conceptually dubious, but in the above discussion we are already taking for granted the power-set operation. Nevertheless this raises the question:
Does the answer to my question change if at each successor stage we merely add all definable subsets from the previous stage?
Does my question make sense? One motivation was that the consistency of PA can be proven in PRA plus quantifier-free transfinite induction up to $ε_0$, which is incomparable in strength to PA.