Every MOC is different, some the LOA will help and others it might not matter. Each MOC does their interviews differently and what they look for therefore is unique to each MOC. They obviously all want a highly qualified candidate that will go on and represent them well. USNA usually sends the LOAs to MOCs. Highly recommend you contact the MOCs and ask if they have received a copy, if not, go ahead and send them a copy.
As to expediting a waiver, no one knows. It is case by case dependent. I have no idea if the medical officer even sees who has LOAs or not. They may just get a pile of waiver requests that they review. None of us know. What happens for one person, might get the opposite result for another. All you can do at this point is prepare for your interviews, continue to work hard at school, keep in shape, work on plan B and C.