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You could run the argument schematically. For instance, using induction in the metatheory you could show that (1) when $\phi$ is an axiom of ZFC, ZFC proves "if $V_\kappa$ is an inaccessible rank, then $V_\kappa\vDash \phi$"; and (2) when $\psi$ is a consequence of $\phi$, ZFC proves "if $M\vDash \phi$, then $M\vDash \psi$".
Let me address the question as stated first. Why does $\sf CH$ has no determinate provability from any of the axioms we throw at it? This is false. As remarked in the comments. Plenty of axioms prove $\sf CH$ or disprove it. Things like $V=L$ or $\lozenge$ imply $\sf CH$ whereas things like $\sf PFA$ and similar forcing axioms imply its negation (these in ...