I don’t think that we have a big disagreement here. The Friedman-Holy models are certainly not canonical and I agree that a key question is whether there are canonical, fine-structural inner models for large cardinals.
But no unverified hypotheses are needed to create the Friedman-Holy models and they witness the compatibility of arbitrary large cardinals with a key component of fine-structure theory: acceptability. We also force Global Square and I suspect that these models can be built as inner models using “internal consistency” arguments for reverse Easton forcings. (Of course the real motivation for the models was to use ideas of Neeman to show that a degree of supercompactness is a “quasi” lower bound on the consistency of the Proper Forcing Axiom).