It's become clear you need to see a chiropractor and a doctor has given you some names to choose from. You would pick one based on which of these criteria:

I always look for a doctor that is into different treatments, I have my regular doctor but I am always looking for someone who steps outside the box when it comes to unconventional medical methods, you only live once might as well make the most of it. Americans always go for the safe standards, but the rest of the world has been living longer than us on different means of treatments.

A well-established practice might concern me that they were behind the times and functioning on long-standing reputation only;

If highly innovative means unproven technique, chiropractics would not be the place I, as a patient, would want to experiment;

Variety doesn't mean much to me in this field either; there can only be so many techniques that really work;

Learning and teaching gives me the comfort of knowing that they are staying abreast of the latest knowledge and are experienced enough to teach to other professionals in their field. Definitely my choice. I'm surprised how few agree with me.

the most important thing for me in choosing a chiropractor, is that he is good.
the guy I go to didnt have an established practice, he was relatively new . . but he fixed my back, and made my kids better. i love him. he doesnt have the most up to date equipment but he knows what he is doing.
love. him.

I was thinking about your question regarding chiropractors. The answer to your question may depend on who the target is for the message. For example, if a person already uses or has used a chiropractor, then length of time in practice may be the key. If a person has never used a chiropractor, then perhaps being affiliated with a teaching facility might convince him/her of the legitimacy of the doctor.

I would imagine that it would do wonders for chiropractic medicine in general to be approved by the AMA or to be affiliated with hospital organizations.

My father was a surgeon and always took us to doctors who were part of a learning/teaching institution. They would have access to the latest technology and innovation in any field and would have to have oversight. You might not want to be worked on by a supervised student, but I would only pursue a doctor who wasn't isolated. Doctors who teach tend to be more established as well.

I'd say none of the above - I'd be asking for personal recommendations from friends, family and my doc