Programs w/ strong reputations for private practice training

I don't think you will necessarily find a list of programs known for producing private practice docs. Trauma exposure and operative experience are going to be extremely valuable for you. As a private practice surgeon, I think it would be safe to assume you will be taking trauma call and seeing lots of knees/hips. My recommendation would be to find those programs that are known as being strong in their trauma exposure and landing good sports/joints fellowships.

As you know in private practice, research and publications are not going to be important. Your surgical outcomes and bedside manner will matter far more.

Do not be so naive. The vast majority of training programs will prepare you well for private practice. If you can not do "standard" trauma and primary joints as a chief, your program has failed you. The connections established at some of the major "academic" programs will open doors for you that would not be open otherwise. Remember, the majority of graduates go in to private practice. Some of the "elite" private jobs go to guys from the "elite" fellowships, and those spots go to residents with connections. Go to a program where you will be happy and succeed. The rest will fall in line. That said, if you are interested in a Kaiser job, try to go to a program that will get you exposure to Kaiser, seems to help from my experience. FWIW, I went to one of the programs considered "academic" here and very few of us are in academics now. Good luck to all.
Also, since I have bothered to login, please remember to take everything here with a grain of salt. Some MS4s post as if they are in the AAOS presidential line when they actually know very little and the reviews of the programs they spent half a day interviewing at are far from accurate.