I wanna say that I think this is a really good response and I am enjoying thinking about this - so in case tone is obscured by the medium, my ongoing engagement is because I find this to be fascinating, rather than being invested in winning an argument. I think you are explaining well why in theory academic medical centers offer training advantages over hospitals that are more solidly focused on clinical missions. My perspective is as someone who trained at a very ivory tower place for residency and fellowship, and although I found it to be a tremendously rich experience by virtue of my interactions with researchers, the ability to engage in the broader academic community of the university, and the ability to connect with faculty who were at the cutting edge of creating new knowledge in the field, I feel like there is an 'emperors new clothes' reality as it relates to the quality of the clinical care at many of these places. Without calling out specific programs - the clinical care at some of the famous 'specialty clinics' that are unique to medicine was really quite poor - an intelligent but inexperienced fellow would typically do an assessment, staff it with a distracted young assistant professor, and then there would be some veneer of sophistication given by a full professor opining about some detail that was seldom critical to the case. At a community hospital down the street, an equally intelligent but less grandiose treatment team was doing about quadruple the clinical volume, and despite the lack of prestige which came from lack of connection to an academic brand and research infrastructure, they did a much better job. The best place to get child and adolescent psychiatric treatment in Connecticut is at a place you have never heard of called the Clifford Beers Clinic. If I was going to be admitted to an inpatient unit in Connecticut it would be Silver Hills Hospital, which is not affiliated with any university but is superbly run. As an attending at a busy community hospital I am struck by how much better the notes and clinical decisions are of my colleagues who are clinicians through-and-through when compared to some of the attending at more prestigious places who just weren't as focused on this part of the role. Things like boarding problems are not unsolvable, they just aren't problems that Academic Medical Centers are optimized to solve. We solved ours here at our humble community system because that is what we do and we do it well. I think the ecology of all this is far more nuanced than we think and there is clearly excellent care to be had at academic medical centers but there is also excellent care in the community and in many cases it may be better. I certainly appreciate that for medical problems it is common for people to need to, and benefit from, visiting specialty clinics and that makes a lot of sense. In psychiatry I tend to think most of these specialty clinics (with the notable exception of dementia) are a lot of faff and when families fly to Boston to talk to someone about PANDAS it usually does more harm than good.
I find it really hard to generalize academic vs community to quality of training because it really depends on what a candidate is looking to gain from a fellowship program to help their long-term career goals. If they're looking for C/L and rare cases, then I think it depends on 1) the quality of the pediatricians & pediatric subspecialists and 2) the clinical acumen of the psychiatry attendings on service, which varies widely even within the same department. A highly ranked Children's Hospital by USN&WR may not have the same level of clinical care for outpatient psychiatry. If the goal is private practice, then the connection to alumni or adjunct/volunteer faculty who have private practices can be highly valuable as they can serve as important mentors and referral sources to getting your own practice set up. Brand name programs can be more marketable for higher prices in PP if you're cash only as higher paying clientele care about stuff like that.
Also, I question whether the quality of clinical care delivered to patients and how well run the administration is translates to the quality of clinical training by the fellowship program. You can have a terrific fellowship program in an unranked hospital/clinic and you can have abusive fellowship training in the top academic centers, the latter of which seems more common from my perception although I have no data. I place higher value on a program's call schedule, the clinical volume, the dedicated time for supervision and didactics, the desirability of the location, the number and quality of attendings (some fellowship programs have only 2-4 child psych attendings total), and the number of specialty clinics and hospital services.
The unique programs are Mayo Clinic and Cleveland Clinic as they are medical training establishments not connected to a university setting so their research tends to be less basic science and more clinical and translational. I'm not sure how much you can extrapolate between a top academic center and training programs in these hospitals/clinics.
If OP is interested in pediatric OCD, I recommend asking specifically about training in this: how much they learn ERP/CBT, using medication for this, working with comorbidities such as Tourette's/Tic Disorders (I can't believe some of my cofellows graduated having never treated this during their fellowship), have an IOP/PHP program for OCD, doing family therapy for OCD, even working with more severe or treatment-resistant OCD.
After going through the fellowship interview trail myself, I found it really disheartening that many fellows graduate receiving no training in diagnosis and treatment of autism, how to diagnose/treat eating disorders, recognizing and managing first-episode psychosis, anything related to adolescent addiction, forensic or juvenile justice cases, seeing young children at all, or doing basic family therapy. From my training, I found it more important to learn how to work with psychiatric disorders that are well established and have commonly accepted treatments (e.g., FDA approved medications or widely used psychotherapies) rather than arguing about the validity of PANS/PANDAS or figuring out if the pt has autoimmune encephalitis.