And how do you, as a first-time patient, assess the quality of psych* you are seeing?

One thing you can do to become a tiny(!) bit more informed is take some psych classes yourself. PSYC1030 is a popular (free) one and gives you the first glimpses into social, clinical and developmental psychology. It doesn’t give you any particular qualification but it does at least demystify some of the processes. That makes it easier to recognise the hucksters

If you’re focused in particular on getting a diagnosis for something, I’d also recommend reading relevant parts of “Psychopathology and Mental Distress” which is a tomb of a book that aggregates both DSM and ICD diagnostic criteria, but also provides current meta surveys on relevant research, from multiple perspectives. It’s reasonably easy to read but definitely better after getting the basic thinking tools from a proper course on clinical psych.

You enter with an open mind, try their process for a bit, and see if you feel better. You can even have conversations with them about the timeline they expect for improvement or what they think can actually be achieved -- "goals of care" isn't just for terminal patients. You can be aware of red-flags, if something feels wrong or misaligned you can discuss it or try someone else, but at the end of the day, you can't know for sure if a modality or a relationship will work for you if you don't try it, and give it some time.

I mean, it's the same question as "How do I assess whether my plumber or mechanic or electrician knows what they're doing." Sure, sometimes there's an easy, obvious difference, but you can't really know for sure whether or not your electrician is grounding your entire house to your hot-water pipes unless you have enough skill and expertise not to need one. After a point, working with any professional requires a weird blend of faith and of assessment you don't have the expertise to actually do.