I want to understand the viewpoint that existence of canonical inner model for a large cardinal notion is strong evidence for its consistency. For example, below is Trevor Wilson's answer to What "forces" us to accept large cardinal axioms?:

Second, there is "fine structure" which gives canonical models for the smaller large cardinal axioms (so far, up to Woodin cardinals and a bit further.) It seems reasonable to expect that a systematic study of the structure of the models of a theory would eventually reveal the inconsistency of the theory if it were inconsistent, and this has not happened yet.

Andrés E. Caicedo's answer to Arguments against large cardinals:

The point of the inner model program (and of its most recent offspring, descriptive inner model theory) is to develop fine structural ("$L$ like") models for large cardinals. These models are canonical in several precise ways, and have a rich internal structure that many set theorists take as evidence of the consistency of the large cardinals under consideration.

Stefan Geschke's answer to the same question:

There is the so-called inner model program where one assumes the existence of a certain large cardinal and tries to build an (easily controllable) smallest model of set theory in which there is such a large cardinal and which contains all the ordinals. The idea is that because we have a good understanding of the final inner model, we would notice during the construction of the model if there were any problems with the consistency of the large cardinal in question.

Can someone elaborate on why the rich structure of the canonical inner model provides evidence for consistency of the corresponding large cardinal? Below are some more precise questions:

Say $U$ is a $\kappa$-complete normal measure on $\kappa$. What exactly does "fine structure" of the model $L[U]$ refer to? I am quite ignorant in inner model theory (maybe I will be more motivated to learn it if I get a satisfactory answer) so below are just words that I've seen here and there: the core model $K$ is the union of all mice, and $L[U]$ exists iff there is a nontrivial elementary embedding from $K$ into itself, and this gives a definition of $L[U]$ that "approximates from below", in contrast to the definition using relative constructibility. Is this what "fine structure" means here?

But then we need to believe in mice in the first place. For example, how to believe in the existence/consistency of $0^\sharp$? Of course it follows from analytic determinacy, but I don't know enough examples of analytic sets to convince myself that analytic determinacy is obviously true...Is there any other good reason?

Can we say the fine structure of $L$ is a strong evidence for consistency of ZFC?

term modelmodulo a canonical system of ordinals. There is no way to construct a term model in general, but it is possible if we rely on $L$ in which every set iscanonical.We may understand the inner model theory program as a project providing aterm modelfor large cardinal axioms modulo some minimal notions (like, ordinals.) $\endgroup$