Well, there are no doctors on this site. We're just folks like you. So we can't diagnose. But it seems criminal to not test him for covid. The symptoms for covid run all over the place, from just a sore throat to just losing sense of smell and taste to all over the map. It would seem in the middle of a pandemic you would at least look at that first. Also, cut out the milk. Dairy can cause inflammation and allergies in most humans. Get a different doctor and get your child tested. If it's not covid, don't now what it is. Could be so many things. Could be allergies. I can tell you that some of the drugs you're getting prescribed are of dubious value without knowing what is wrong, and some of them could be making it worse by lowering the immune system. One of them has already proven to be not useful if it is covid and might make it worse. Get a different doctor. If you need to, go to a better facility, such as a university hospital, if you can afford it. And yes, if you can find the best pediatric physician there, that doc would probably test for covid.
I agree that he should be tested for Covid-19. It has been shown to take different forms in different people, and to last a long time in some people.
I would also ask the doctor for a chest x-ray for your son. He might have inhaled something that is lodged in his lung.
For my son this went on for a few years of coughing so much it made me crazy. Taking so much antibiotics. Age 4-6. Yes is was a sinus infection. It kept coming back. One day I finally got him to do the neilmed sinus wash. I did it first show him it sucks but its funny. After the third time he used it, a huge nasty clump came out. End of his problems. He is 14 now. Thank god we finally tried it. Good luck. Hope he gets better quick.