Personally I do not believe that a good teacher makes a successful student but it depends. If the student relies on a teacher for everything then they need a good teacher. I had many bad teachers through my childhood who even made me cry but did I give up the piano? I learnt the piano on my own terms not anyone else. Even when I studied with some of the worlds greatest pianists I felt what they could only provide was the networking in the musical industry. Bad teachers teach you a great deal as well as good teachers. You do not always have to have positive experiences in life to learn a lot often we learn a great deal more from negative experiences but it is our own fault if we live in negative experiences and not get ourselves away from it and improve. Some people experience something negative and let it get them down, do not know how to use it to inspire, some do not realize that they are in a negative situation, like a frog in water that slowly heats up.
There is really no magical golden rod of knowledge that a teacher can transfer to their students although some of us would like to imagine there is! There is direction but there is no magical instant change instruction, you must take a concept and work hard with it yourself and use your teacher as a guide, the teacher cannot do the work for you nor can they motivate you to be consistent. I motivate all my students during lessons but I can be certain that this inspired feeling to do work vanishes pretty fast as soon as they are left to their own devices. Self motivation comes not from a teacher (you may learn about its existence and what tools we can use to motivate ourselves) but from yourself and how you deal with life as a whole. Self motivation and discipline is what makes you go far, you can spend a million dollars a year on piano lessons from the best teachers in the world but you will get absolutely no where special unless you actually work on it yourself by yourself.
In the end the student must want to do well and work hard at home the small amount of time with a teacher every week is hardly enough to contribute a large percent to the overall success of your musical journey. I do believe that you can have a bad teacher and not realize it, and also not realize what is best to learn at the piano for yourself or what interests you. I believe that a teacher should be used as a tool to improve the rate of your learning not necessarily as the basis of your learning. People have to start exploring music themselves, make decisions themselves then use their teacher as a compass to ensure that they are still on the right track. Unfortunately most people treat their teacher as the sole source for inspiration and direction in their music which sets them up for troubles (this is ok for a child but when does the teacher encourage them to make their own decisions and acquire a musical taste? A teacher who allows their student to become reliant on them for direction is setting up their student for trouble and insecurity).
In the art world we must walk it ourselves and enjoy it ourselves, then teachers or people who have taken the musical journey much further than yourself can direct and suggest what you should do. I retract in horror for the student who relies completely on their teacher, usually these students do not last long since their motivations do not come from themselves but are created from other peoples expectations. If I meet someone who relies on everything I say I often get them thinking on their own during lessons, when I have students who think they know it all and "do not know that they do not know" I will take them aside and suggest direction, there is always this opposing force you must work against.
For a childs mind a great teacher is very much required. A child can be emotionally crushed by bad teachers or bored to tears and hate music. Often I have had new young children who came from other teachers and find that they have learned only early pieces which sounds like C major scales

. These children have even said to me that classical music is boring because of this! They only have this small window and their previous teacher never revealed to them the wonderful, mysterious and various world of music out there.
So for the young mind the best teacher should be sought (if you know that they are very interested in the piano and it is not just a passing curiosity), for adults we can put up with bad teachers (take the knowledge that works for us and discard the others) and learn to move on, but children need to be carefully nurtured. A child can feel trapped and frightened with a bad teacher and adult merely can change teacher.
For adults usually if you socially connect with your teacher then this teacher will set up a good learning environment. Of course what they teach you must work as well, but I find if a teacher treats an adult like a child merely directing and correcting and acting as an authoritative figure it can belittle the adult student and no matter how good the teacher is the adult student will not be in a good learning environment mentally. Of course if you pay $500 a lesson you may not want niceties and merely want to learn a great deal every lesson but knowledge can be wonderfully transmitted when you feel it comes from yourself and comes with great gentleness and clarity, rather than being rammed into you and forcefully changing and acting against your current method. The best teacher can transform your playing without you realizing it most of the times, they can shape what you have, they do not try to recreate you but enhance your very own two hands with personal consideration to your method. If you pay good money for lessons you should expect this. Any teacher can teach the model of perfection but it takes a great teacher to deal with the tools that the students currently has.
A teacher who does not know how to socialize with their adult student should not teach adults, the teaching style should be friendly as such but with periods of serious consideration, in the end you need to be able to laugh at each other as you solve your own problems I have found this to work best with adults and not such much a sterile clinical approach.
A teacher needs to maintain a good rapport with their adult student. This is of course essential for children as well but I have found that some young children do not require it so much as adults do. An adult want to make friends with their teacher generally where a child needs to respect their teacher but also know they can have fun with them.