Is your intention to try to break into research in a serious way? Then 100% go academic, as prestigious as possible. Do you just want to be involved in academic psychiatry long-term? Then academic will make it easier, but not strictly required. Do you want to set up a high-end private practice in a saturated city? Then the more fancy names you can associated yourself with the better.
Once you descend from the Olympian heights, it makes very little difference to your career prospects.
I am biased in favor of academic programs provided they are real-deal tertiary referral centers just because you get a chance to see weirder stuff and more complicated cases, but this is a question of being better at dealing with maybe 5% of the patients you will encounter in the real world, not the vast majority.
I really like the way you're breaking it down, and agree with the importance of ensuring an academic program has a really robust tertiary referral center if you're planning on practicing in the community. High-end practices in a saturated city are in some respects a tougher market to break into / require more diligent advertising and referral management compared to an underserved urban, suburban, or rural setting.
I will say though that it depends on what the experience is. Please, if I'm wrong on any of this CW or others, call it out because I think a discussion point-counterpoint from us and other members would be helpful for Osminog and other medical students trying to come to a similar decision.
I imagine CW had a great experience with some of the rarer or more complex cases where they trained, though I am venturing to guess that there were still fewer and/or less severely treatment-refractory psychosis cases than, for example, my training that involved 12 months of state hospital level care (a different form of tertiary referral center).
CW's program probably was rather academic in the approach and systematically applied very good algorithms and very up-to-date treatments. The patients were still probably not as violent, agitated, and profoundly ill as the ones I trained with. It probably leads to me being very comfortable in a state hospital, correctional, or underfunded community program whereas I imagine CW might occasionally miss the highly structured specialty center when it comes to managing those patients and I don't know what I'm missing out on in regards to that.
This being said, my program may have been unique among non-affiliated programs in the fascinating breadth of opportunities. 12 months of state hospital care, 4 months of private-for-profit inpatient, 3 months of VA, then a combination of outpatient clinics with medicaid patients, cash-only patients, private insurance patients, student health center patients, and 2 years of child/adolescent outpatient clinic. At least one resident per year was doing research at the NIMH throughout training and fast-tracked into a fellowship through the NIH. Of course, we would have had to do an elective to have HIV specialty clinic (though with the rate of HIV so high in our population, it might not have really mattered), bipolar specialty clinics, etc.
Adding to my previous post, if a prospective trainee wants a particular experience that a program doesn't currently offer, it's important to look into what it would be like to gain that experience at a given program. For example, if one wants to learn psychedelic psychotherapy or work in an IV ketamine clinic for 2-3 months while in residency, it might be a different process at an academic program that is currently conducting psychedelic research and has an IV ketamine clinic than it would be at a program that offers neither but is open to residents forging the path for an elective in those fields. There are merits to putting some of that onus onto the resident, as it offers a practical experience in managing negotiations between bodies, developing a training curriculum, and searching through clinics to determine which would be the most beneficial / easier / harder to work with. There are, of course, all the frustrations associated with that which might not be desirable if someone intends to simply sign a contract to work with GreenBrook or a similar company, working in an academic ketamine clinic, partnering with an anesthesiologist, joining a group, or going into it as a solo provider when they graduate. It can be a lot to juggle while in training, but there's no time like training for getting your feet wet with something that complex (again, this is showing my own biases).