I asked a similar question on SE before and received an answer. Not completely convinced, I decided to ask it here with some modifications. Note: I understand how forcing works and how it proves relative consistency statement in the metatheory; this question is more about the philosophical interpretation, hence the tag "soft-question".

There are mainly two approaches to forcing as far as I know: the internal approach, usually with a complete Boolean algebra, and the external approach, using a countable transitive model (ctm). While the internal approach is elegant and intuitive in my opinion, it only creates a Boolean-valued model instead of a usual transitive model. Moreover my impression is that this is not how people usually think about forcing; I feel that when people do forcing they really imagine that some new set is created out of thin air and thrown into the current universe.

Now for the ctm approach. The problem (I feel) is that it only creates a ctm. All kinds of stange stuff can happen for ctm:

1. (Hamkins) A ctm can be pointwise definable.
2. (Hamkins) Every ctm embeds into its own constructible universe.
3. There exists a statement $\phi$ such that ZFC+"there is a countable transitive model of ZFC$+\phi$" is consistent (relative to some reasonable hypothesis), while it becomes inconsistent if the word countable is changed to uncountable. An example of $\phi$ is "there does not exist ctm of ZFC" (consider the minimal model). Is there a more interesting example?

Basically, my question is why should one believe that forcing is meaningful? Yes forcing shows there is a ctm of $\lnot$CH, so $\lnot$CH is consistent, but could that be merely a pathology about ctm, just like 1-3 above? I mean there is a model of ZFC+$\lnot$Con(ZFC), but it has to be ill-founded so nobody believes this should actually be the case; I don't feel much better about ctm than ill-founded model. For the particular case of $\lnot$CH, another doubt of mine is that the von-Neumann hierarchy picture is often used to justify the consistency of ZFC, but the forcing argument seems to say that we actually never reach the "true" power set of a set, so the hierarchy cannot proceed in the first place. Of course large cardinals directly imply the consistency of $\lnot$CH, but that is another story.

Here is an even vaguer question: set theory people often start a sentence with "collapse $X$ to a countable set". If we take this seriously, does it mean actually all sets are countable, and uncountability is just an illusion? Dana Scott seems to share a similar view in his foreword to the book Set Theory: Boolean-Valued Models and Independence Proofs:

......Perhaps we would be pushed in the end to say that all sets are countable (and that the continuum is not even a set) when at last all cardinals are absolutely destroyed. But really pleasant axioms have not been produced......