Why can't we prove consistency of ZFC like we can for PA?

The problem is that, unlike the case for PA, essentially all accepted mathematical reasoning can be formalized in ZFC. Any proof of the consistency of ZFC must come from a system that is stronger (at least in some ways), so we must go outside ZFC-formalizable mathematics, which is most of mathmatics. This is just like how we go outside of PA-formalizable mathematics to prove the consistency of PA (say, working in ZFC), except that PA-formalizable mathematics is a much smaller and relatively uncontroversial subset of mathematics. Thus it is a common view that the consistency of PA is a settled fact whereas the consistency of ZFC is conjectural. (As I mentioned in a comment, as a gross oversimplification, “settled mathematical truth (amongst mainstream classical mathematicians)” roughly corresponds to “provable in ZFC”... Con PA is whereas Con ZFC is not.)

As for proving the consistency of ZFC, as you mention, we could go to a stronger system where there is an axiom giving the existence of inaccessible cardinals. Working in Morse-Kelley is another possibility. In any case, you are in the same position with the stronger system as you were with ZFC: you cannot use it to prove its own consistency, and since you’ve made stronger assumptions, you’re at a higher “risk” of inconsistency.

Using ZFC plus the axiom "Uncountable strong inaccessible cardinals exist" to give a model of ZFC is exactly the same kind of "stepping outside" as when you use ZF to make a model of PA.

Of course we can. And we do just that.

To "step outside of $\sf PA$" means that you assume the consistency of a far stronger theory, e.g. $\sf ZFC$, which lets you construct $\Bbb N$ as an object and prove it satisfies $\sf PA$.

When you say "assume that $\kappa$ is a cardinal such that $V_\kappa$ is a model of $\sf ZFC$" you effectively saying "We are stepping outside of $\sf ZFC$ into a theory "$\sf ZFC+\varphi$ for a suitable axiom $\varphi$, and there we can can find a model of $\sf ZFC$.

In fact, when you assume something like $V_\kappa$ is a model of $\sf ZFC$, you can even find a fairly canonical model of $\sf ZFC$: $L_\alpha$ for the least $\alpha$ satisfying $\sf ZFC$, where $L_\alpha$ is the $\alpha$th step in the constructible hierarchy. As to why it exists, that's another question. But it is there.