Preface: I somewhat wanted to post a comment vs. an answer, but this is obviously far too long. Take it how you want. I've got some sources, but not what you were really looking for. The tl;dr summary is that:
- I concur with the compartmentalization theory, and think we all have our irrationalities
- Religiosity may be primarily influenced by genetics
- My personal experience as a deconvert surrounded by religious relationships supports the compartmentalization theory
- I see less critical thinking issues in religions where no testable claims even exist and thus think critical thinking development is not hindered outside the religion itself... but wonder about religions/pseudoscience indoctrination (like homoepathy) where the foundations of critical thinking itself (like empirical evidence) are undermined.
I think you're on the right track when you bring up compartmentalism.
Here's some sources suggesting a connection with genetic inheritance when it comes to religiosity (thus critical thinker or not... one might be a bit more predisposed toward religious/spiritual beliefs):
Why do I bring this up? Because I think the causation lines (if any) may run the opposite direction. I would not be surprised to learn that critical thinking/analytical tendencies are correlated with lack of religion such that the former implies the latter vs. the latter preventing the development of the former.
It seems that most of us have compartmentalized our skepticism to some extent. The religious folks just have an easily identified one. One which safeguards itself from scrutiny.
This may be the exact case. I don't know if you're familiar with Eliezer Yudkowsky and the rationality-dedicated site he created, LessWrong, but HERE is a discussion of this exact quote. The author points out that theism has become the one fairly agreed upon example of irrationality (non-critical thinking) and questions whether this should be the case. Not a great source, but just thought I'd point you in the direction of others like yourself asking this question.
Now I diverge into experience/background. I deconverted from devout, wholehearted, fervently lived Catholicism about 18mos ago (full story if you really want it HERE). What I wrote above does track with my experience and my nature. I have always been analytical and curious. I'm an engineer and took things apart as a kid. I want to know everything by the time I die. I analyzed and analyzed before I could buy something because I had to have the best purchase supported by the evidence (specs, user reviews, magazine reviews, etc.). I fact-checked claims I heard. I made spreadsheets to make sure I had the finances to buy a laptop when I was in college.
But, I never turned my critical gaze toward religion. I literally just thought it was correct (and that was supported by a conversion experience I had) and started from that point -- in other words, my intellectual investment was in theology and spirituality, not apologetics our the foundations for why to believe.
One day I wondered if non-gospel writers had written about Jesus and was sorely disappointed by my research results compared to what I expected. That was all it took for me to pursue my religion just like all my other endeavors. I suspected that the best way to analyze it's truth was to suspect its falsehood and try to prove it back to myself. I've failed in these past 18mos and am a non-believer, but one who's still got some books on my to-do list in order to put the case to rest.
I'm surrounded by believers (7 years of my adult life was spent building friendships and relationships with other Catholics), and my experience has been that many are critical thinkers... but I would almost describe it as them not being free or able to question their own religion when we get into discussions. If I construct a complete analog for another religion or particular diddy from pseudoscience, they will agree wholeheartedly. But it's like there's a complete disconnect when those same critical principles start poking around theism.
So... my answer is that 1) genes probably have a lot to do with it in the first place and 2) my suspicion (yet to be confirmed by studies) is that indoctrination might be more accurately said to create a "shell" around one particular area such that one can be highly critical but still a believer. And 3) we all compartmentalize, but the beliefs just aren't as divisive and so we aren't nearly as aware of it.
I'll end by saying that I think most religions don't have a very good intersection with test-this-claim-now areas of the world; their claims are not predictable, verifiable, testable, reproducible, universally discoverable, etc. Thus, I think I'm gaining leniency with particular forms of theism that don't really matter except in the believer's. Sure, phrases like, "I'll pray for you" may be grating, but many/most will still recommend chemo, counseling, etc. to get "real" treatment. It's like a play belief that they don't really believe.
I'd be much more interested in whether indoctrination with things like homeopathy, crystal healing, etc. prevent critical thinking, for in those ideologies, there seems to be a direct suggestion that evidence isn't really evidence, that all "supposed" scientists are involved in a gigantic conspiracy and don't really know what they're talking about or don't have the "spiritual eyes" to see, etc. In those types of belief, I really could see a hindrance of critical thinking, because they entail a direct opposition to empirical evidence and other foundations of critical thinking in the first place.
For the rest, I think their sources of evidence are generally one sided and many may simply never get the "itch" to research their beliefs -- or a crack in the shell that lets them do so. But, I'm skeptical that such occurrences prevent critical thinking elsewhere.