Unless you already know what kind of specialist you need to go see, and you can afford to go see the right one (might involve multiple visits to multiple specialists), you might literally never get help for a chronic problem that is not obvious on a differential diagnosis.
There really needs to be two kinds visits available: quick visits and check ups, and diagnostic visits. The first can be over in 10 minutes, the latter should last a solid hour.
I'm starting to wonder if "diagnostician" should be a separate specialization. Then I could make an appointment specifically to go see the diagnostician, and not bother the general practitioner, who is evidently too busy to spend more than 15 minutes diagnosing something.
Internal medicine specialists are supposed to be "diagnosticians," as you envision it -- they have more training than general practitioners if they're internal medicine specialists. But they're one of the lower-paid specialties, in part because it's easy to charge for a stent or injection but in the US the act of diagnosing a problem doesn't really have a code.
He also didn't take insurance - because he didn't want to bother with it. And he wanted to be more honest in his notes than his libertarian heart felt comfortable disclosing to the insurance company or the government payers.