As to whether or not an LOA helps with a medical waiver, here is my best guess. Please note, it is only that, NOT official policy.
If you can't participate in plebe summer due to some medical issue (injury or otherwise), it won't be waived. So, having an LOA won't help. This happened to one of my candidates last year -- he wasn't sufficiently recovered from surgery by I-Day. Great candidate, highly recruited, LOA . . . no waiver. He had a great Plan B, BTW.
I suspect that if it is an issue such as color blindness or something else that allows you to participate fully at USNA but may limit your career options down the road . . . well, USNA can only take so many of those individuals. In that case, I would THINK that having an LOA MIGHT help (not be determinative) because it shows you are a highly qualified candidate.
The above said, I still believe the predominating consideration is medical and, thus, the answer to your question may largely depend on your particular DQ. I recommend you do NOT post details here as, even if you told us about your medical condition (which I recommend AGAINST), we still can't help you.
Contact Larry Mullen.