From the progression of your posts, it sounds like your son has had shoulder symptoms for at least a few months, and even with rest(and I assume PT/anti-inflammatories) the symptoms are continuing. What is not clear is whether your son is at complete rest or is in some type of throwing program through July 14. What seems clear is he is not progressing.
An MRI without contrast would not be the best tool. Do you know if the MRI was with or without contrast. I am surprised by your comment the MRI showed "nothing." If there is impingement, it should provide an explanation, i.e., where is the area producing the impingement. If your son has both shoulder impingement and tendonitis, why? What clinical findings lead the doctor that direction as opposed to others? I can tell you from personal experience that impingement and bursitis can be diagnosed when the actual issue is labrum related. With a football player, and the type of incident which can impact the labrum, one has to wonder, at least I would if it were my son.
If your son is not better by July 14, I would not be letting any doctor put a needle into his shoulder without a second opinion from an orthopedist who specializes in shoulders and sports/baseball, such as Dr Andrews. Get on the ASMI site and find your son an ortho in your region which they recommend. College team ortho sounds good but rings hollow to me, at least. I learned that when an MLB team ortho misdiagnosed our son's shoulder as impingement and bursitis, treated it accordingly including the injections, which misdiagnosis cost him his baseball career.
I realize you are following the advice of the doctor and also hoping he is right so your son can be recruited. Everyone on this site wants the best for your son and wants to help you help him get recruited.
From what you have posted, I think you need to make sure the diagnosis is correct. Your son needs another opinion with a contrast MRI done by an ASMI referenced doctor, especially someone like Dr. Andrews himself. If the MRI was with contrast, your son needs a second opinion and a second review of the MRI and you need to get a copy yourself and make sure it shows "nothing." It has to be with contrast, though.
Good luck with getting a better understanding on whether impingement and tendonitis is the correct diagnosis and if so, why it is not improving with rest and hopefully PT and anti-inflammatories.