This question consistently comes up again and again but deserves unpacking, because many students and trainees don't really understand this.
1. In marketing psychology, there is a concept of branding association. Using consumer's positive association/transference to brands rubs off on you by association. So if people think highly of Harvard or Yale, and you prominently report that affiliation, those people may be more disposed to see you in a positive light.
2. There is also the concept of, "If you've got it, flaunt it." Thus it is not surprising that those that went to prestigious institutions are more likely to emphasize this in their marketing and websites than those who did not. This is not the same as "if you don't have it, you're doomed to failure" but that you need to find other things to emphasize to communicate your value if you don't have those credentials to flaunt.
3. In these divided times, respect for ivory tower institutions is not what it once was. Some people now see them as bastions of elitism (which makes you elitist), or factories of "woke indoctrination" (and worry you will poison there children), and some of these institutions have been tarnished by accusations of failing to deal with antisemitism etc. Put simply, branding association is not always positive. Similarly, in some locales, branding association may be stronger for the institution with the college football team than a faraway Ivy.
4. You will find that there are a lot more cash practices from graduates of the top residencies (which doesn't always translate to public prestige). This is not simply because of branding or prestige but actually about the quality of training, the amount of psychotherapy training, exposure to psychiatrists in private practice etc. Ill take these in turn.
5. Cash practices are more likely to do psychotherapy. That's not necessarily because more people want psychiatrists doing therapy, but because insurance doesn't pay as well for therapy. You can make as much or more in an insurance practice on average accepting insurance for med management. For psychotherapy, there is a clear financial advantage of doing cash only. Top programs are more likely to train their residents in psychotherapy. It's also easier to fill you practice if you have therapy patients - 1 therapy patient = 8-48 med management patients.
6. While there isn't a correlation between the "name" of the program and quality of training, the top 20-25 programs do provide generally better quality of training that most other programs. If you do good work, you'll retain patients, and attract more patients due to positive reviews, happy physicians referring their patients to you, and happy pts referring their friends to you.
7. Because top programs are more likely to have graduates in cash practices, you are more likely to have these supervisors in training and network with people whose practices are full and will refer you patients if you attend one of those programs than one where you don't. Again "top ranked" and public "prestige" are not one and the same, but there is something to it. Of course, it's not impossible otherwise, but not as easy. I wouldn't consider my residency program to be "prestigious" but it was a top ranked program and we did have multiple talks on setting up a private practice and resources in the residency office to help with that.
8. The further you are out of training, the less your training will matter and the more your reputation and the work you do matters.
9. If you have no head for business (or no interest in it) and no one to help with that, then your practice won't be as successful as it could be. Honestly, seeing how many psychiatrists in pp have no idea what they are doing was what gave me the confidence to go out on my own. Conversely, if you are business savvy, that is more important than the "name" of your program.
10. Branding Association is not limited to residency training - it also includes college, medical school, graduate school, fellowships, and faculty affiliations.
11. Another marketing principle is FANAFI - find a need and fill it. While there is a clear need for psychiatry (especially CAP), if you are more laser focused in identifying a need and creating your practice and marketing around it, you will be more successful.
12. There might be other things that you wish to flaunt - books you've written, news/TV appearances, invited talks, important publications, important leadership roles, less common board certifications, specific psychotherapy trainings/offerings, awards you've received, clinical experience you've had, personal experience you've had. Prior experience (e.g. prior business experience now focusing on executive coaching, prior tech/start up experience now focusing on founders, former college athletics now focusing on sports psychiatry) etc.
tl;dr - Yes, it matters a little, but the quality of your training matters more. And all of this matter less the further out from training you get. There are multiple things one can potentially flaunt, and thinking carefully about you brand yourself and market your practice, in addition to doing high quality work, and having (or willigness to develop) some business savvy are all important. Ultimately, in private practice, the brand is YOU, so you need to invest in cultivating and maintaining that so people see value in what you offer.