I can empathize with your frustration. Several years ago, my wife and I were sick with high fevers from sinus infections while at Disney with our 2 small (at the time) children. We were staying at the Polynesian. We were so sick we took turns sleeping all day while the other parent forced themselves out into the parks with the kids. Disney was *zero* help, except to provide numbers for the local walk-in clinic, hospital, and concierge doctor. Since we had no car (and I'm not sure if Uber existed back then), by day 3 we gave in and called the concierge doctor. A guy showed up at $300+ a pop ($600+ total) to tell us yep, we have sinus infections and we need an antibiotic, which thankfully he had with him at some inflated price. Prior to calling the concierge doctor, I called the local walk-in clinic and was told it was a 1.5 hour wait and would be something crazy like $350 to be seen. Since both of us were sick, we weren't about to hop in a cab with our two kids and waste a day doing that for more money than the concierge was going to charge. Of course, we started on the antibiotic the concierge doand were starting to feel better right about the time that we were on our way home.
The year after that, we were again on property and my daughter ended up with a sinus infection. We knew it was a sinus infection because she always got them after a bad cold. The doctor on call at our pediatrician's office refused to treat her without her being seen, and he also refused to let us speak with our own pediatrician--but that was his mistake, because our pediatrician was the practice owner (who apologized profusely once we got home), and this particular on-call doctor disappeared from the practice shortly thereafter.
In my personal opinion, the whole setup is a racket and Disney is complicit in it. I agree that it's not Disney's responsibility to provide medical care, but for most people it's difficult to cancel or reschedule trips to Disney, and they really should help to facilitate care--at a reasonable rate--for families who end up sick during their resort stay. Instead, all they do is refer you to local providers who are more than willing to take advantage of your unfortunate predicament.
There are now "virtual" medical providers which will usually do a video conference with you through an app or online, and are willing to treat minor illnesses. I believe in most cases your employer or medical insurance company must have a relationship with one of the providers for you to use them. About a year ago, I tried one such provider for my daughter while at home, and was able to obtain an antibiotic to successfully treat her ear infection, for a much reduced rate (I think it was $45-$50 for the consultation). Strep is a bit tougher since the symptoms can mimic other illnesses, and without a strep test a physician may not prescribe. If available to you, I'd still give it a shot before I'd shell out hundreds of dollars to a "tourist doctor."