Regarding the rarety of my condition, it's diabetes treated with insulin, so not rare by any stretch of the imagination. With a positive diagnosis and in a fairly uncomplicated case, it's a huge burden on me to learn just the single condition. Now imagine doctors who don't have to deal with only a single, well-diagnosed condition.
You say that a "sprained ankle" is easy to diagnose -- but is it really? On the differential, how many other conditions manifest similarly? What complications can a trained doctor see, what questions do they know to ask?
In my interaction with the medical industry, in no case would I have been as well-served by a "lesser expert."
Regarding the term rent-seeking. This is quite a specific term, that means causing payment to be made for something that would be free. For example, if I opened an institute to monitor air quality and make recommendations, and managed to get the government to force everyone to pay an "air tax" for my endeavors -- this is "rent-seeking", since, of course, air can be enjoyed without paying for it.
From how I read your comment, you did not intend this usage. Rather, you meant to speak about the supply of trained doctors (not to speak about something that might be free otherwise), so it is more cartel-like behavior, at least as far as I read your argument.
I assume you still want the lesser experts we're discussing to be regulated legally, etc, and weren't comparing with "free" opinions such as if I ask an aunt about herbal medicines that have worked for her. If the latter, then, sure, I would say it is rent-seeking to start charging people to discuss herbal medicines for which discussion is otherwise free.