I’m not sure how helpful this will be, but I wanted to give you my perspective. I had three piano teachers as a kid (we moved around a lot) and a couple as an adult re-starter. All of these teachers had quite different personalities and teaching styles, although I can also appreciate certain fundamental similarities in terms of thinking about how to teach piano. Anyways, despite all of them being different, I do think that all of them were/are very good teachers, and I learned a lot from all of them. I am not someone who’s ever really auditioned teachers. The only time I ever decided that a teaching situation wasn’t working for me was with the person I started with as an adult, and that was because his availability significantly decreased, and I wanted more help/support than he could provide.
I have not sought a very particular list of things in a piano teacher. That said, all of my teachers were/are good pianists and have had good training. I’ve overall tried to always be very open to what they feel they have to teach me at the piano. When I re-started as an adult, I was very willing to do lots of exercises and such, but both the people I’ve had as an adult were not excited to spend their time with me guiding me through exercises and encouraged me to bring what I wanted to play. Again, I would have been very happy to do those exercises (and kind of had expected they would want me to do those, too), but neither of them was keen. So I don’t play exercises.
My current teacher is a Taubman person. I didn’t start with him because I had an injury or was really keen to learn what Taubman was about. But he was recommended as a good teacher. It was clear from my first lesson that what he really was passionate about was the Taubman approach. He didn’t insist on a full re-training, but we have done a lot of Taubman work, and all my lessons are very much steeped in the language of Taubman. I have found this to be very helpful. I also know that if I was reluctant/not willing to be open-minded about things Taubman, it would have been very hard for me to be happy with this teacher, and I think I would have been struggling with him and myself throughout any relationship we had. Instead, I have learned some pieces with him that were ‘bucket list’ pieces for me — pieces that I really wanted to play but had challenges that I couldn’t solve on my own and hadn’t been able to solve with any other teachers.
I guess what I’m trying to say is don’t let the perfect (or what you think is perfect) be the enemy of the good. Try to be openminded and meet your teachers half way, exploring, investing, and believing in them and what they can offer you. I think there are lots of teachers who have much more to offer than what we can likely take full advantage of.