This is a tough problem! Writing this to address your concerns but also for the comment readers. I don't provide great solutions, but I think validating your problems is helpful and hope there's something useful in here:

I've been thinking for a while now that we need people you go to help you find a proper therapist. Like a consultant/dispatcher/human load-balancer/Mental Health GP. Medical GP's have no idea what they're talking about in this department (save some outliers) and would probably admit to that if pressed.

The approaches as they stand now aren't ideal, so I feel your frustration:

1) Pick one that accepts your insurance located near you: Those lists that the insurance companies publish aren't up to date and Psychology Today is also dicey. And you're right, they all start to look the same (they are most definitely NOT in reality). Many mislead by saying they accept insurance but really mean out-of-network benefits, which typically aren't great especially if you're not working for a mega-corp. There's also a sizable body of therapists who will charge high rates (not in proportion to quality) because some rich clients won't care. It's a bit dirty to me, but a lot of the time they'll reduce their rate very quickly and dramatically once they know you're not rolling in hereditary wealth. For example, I've seen $350/hr off the bat reduced to $100/hr with one simple statement of "that's outside my budget".

2) Pick by approach: In theory is a good way to start, because some modalities probably aren't a good fit for certain classes of problems and personalities. However, it's a tough balancing act. A therapist who is too focused on one modality is probably a bit inexperienced, because they're in everything needs a hammer mode. However, listing too many modalities sometimes indicates a lack of rigor, focus, point of view.

3) Sites like Therapick are great in theory in that you get a video introduction to assess someone who might be a good fit. However, they tend to attract less experienced therapists (who could be good!) who are trying to build up their client-base. Just like rent-stabilized apartments in New York City, you're less likely to see the most experienced and popular therapists doing active recruitment like this because they're snatched up quickly and don't need to spend the time and effort.

4) I would highly discourage any of those phone/texting therapy services. Who knows, they may work for some, but they don't strike me as plausible solutions for most, if not all, people. If it's money that's a concern you're trying to solve for (which it usually is to some degree), there are other approaches to solving that. If time and convenience is a concern, you really need to get reevaluate what you think therapy is. To me, these services are bandaids, which I suppose have their utility.

5) Recommendation through another therapist: This can work, but you should be aware that therapists form milieus — talented ones tend to hang out and consult with talented ones, and vice versa. So if you have a bad therapist, it's likely their reference is as well. My personal, albeit possibly offensive, theory is that there's small pool of truly gifted therapists (less than 20%). I subscribe less to the notion that there's a perfect match therapist for everyone and that it's just a matter or style and personality.

On top of all this is what I called the Fundamental Therapy Problem in that it's quite possible you shouldn't be the one deciding what style would be best for your long-term interests. Don't think too hard about this one, because it'll drive you nuts :) But it helps to try to assess based on gut feelings of feeling secure and feeling challenged, rather than a heady, logical assessment.

At the end of the day, it's common knowledge (and backed by some studies) that it's really about the relationship between the people. If you feel intimidated, baffled, weirded out, or annoyed consistently — or if you don't truly respect your therapist — it simply won't work. Don't waste your time hoping to get used to it, it won't get better.

In your situation, I'd recommend a strategy of finding a good-enough therapist to start. But there'll be a point where it'd really help to find an excellent one. That will be easier once you've leveled out a bit.

In conclusion, references are the best from friends you trust (hard to get sometimes because of privacy concerns), and then a scattershot (guess and check) approach. Expect it to be a bit of an expenditure when you're interviewing around at the beginning. But have faith, things can get better rather quickly once you have a decent therapist situation established.