I could not find a question that seemed to answer my specific query, despite lots of material on the Continuum Hypothesis (CH) on MSE and MO. If there is already a question on this, I'd greatly appreciate a link.
My question has to do with the resilience of CH to being resolved. I realize that it is independent of ZFC and "most" large cardinal axioms, and that CH can fail as "badly as you want" (see this question).
My question is: why is this the case, and is there intuition as to why "ZFC + any large cardinal axiom" cannot seem to anchor this problem? I realize the "multiverse" approach is simply to study CH in whatever universe you are in. But my question whether there is intuition as to why CH has such an indeterminate status no matter what axioms you throw at it. Thanks!
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 particular imply that $2^{\aleph_0}=\aleph_2$ of all values).
Specifically you might be asking about large cardinals. Well, this is because large cardinals are large, and the continuum is small. Historically, I believe, the motivation for the thought that large cardinals might settle the continuum hypothesis came from the fact that we can prove the continuum hypothesis for Borel sets. Namely every uncountable Borel set has size continuum, and we can push this to analytic sets, but not further.
However, if there exists a measurable cardinal then we can prove the continuum hypothesis for co-analytic sets as well. Which is a push forward. So perhaps by having "enough" large cardinals, or a strong enough large cardinal axiom we can push this proof and go through all the possible sets of reals?
Well, no, not really. While these things definitely can occur (in the presence of Woodin cardinals we can push the continuum hypothesis higher up the projective hierarchy), there are more sets of real numbers than that. And at some point the large cardinals exhaust their power.
In particular we have the Levy-Solovay phenomenon:
Namely, small forcings cannot change the largeness of an inaccessible cardinal. The same is true for many other types of large cardinals, e.g. weakly compact cardinals, measurable cardinals, Woodin cardinals, etc.
On the other hand we always have a forcing of size $\aleph_2$ which violates the continuum hypothesis: adding $\aleph_2$ Cohen reals; and a forcing of size continuum which restores the continuum hypothesis: Levy collapse of $2^{\aleph_0}$ to $\aleph_1$ (and it doesn't add real numbers too!)
So with particularly small forcings we can switch the truth value of the continuum hypothesis on and off as we like. Therefore large cardinals, which are unaffected by small forcings, cannot possibly decide the truth value of $\sf CH$.
(That been said, there are large cardinals properties which can be destroyed by small forcings, but so far these were shown consistent with $\sf CH$ and with its negation by various means, e.g. by showing that we can perhaps destroy the large property, but then we can have another forcing which restores it and doesn't change the value of the continuum.)