Nobody cares. They'll go about their day and think about other things. I'm sure if you see enough rheumatologists, somebody will diagnose you with an autoimmune disease. You can always find that 1/10 dissenting opinion if you see enough of them. Same with any other common disease/complaint in another subspecialty (IBD in GI, heart palpitations in cardiology, etc). See enough of them, you'll get a diagnosis attached to your medical record.It's up to you decide if all the other doctors were just lazy/stupid and if that final rheumatologist you found that agrees with your self diagnosis is actually correct.
You will have to wait a bit as a new patient but that is normal. I referred someone with MS to Mayo's clinic and they did an amazing workup--that person goes there annually now for follow up. I would start by going online to their website for JH and reach out, see what conditions they specialize in. https://www.hopkinsrheumatology.org/make-an-appointment/ I don't have experience with which centers in the country are considered the best for your condition but you can definitely research that too. Any place where you make an appointment will let you know where to have your past records sent, however be prepared to have labs and tests repeated for thoroughness.