Have they done an MRAngiography yet? sometimes an MRA will show what is causing the blood flow to stop and sometimes only when they go inside they can see the cause like in my case, there were fibrous bands compressing and kinking my subclavian artery (left side) and bands are hard to see in imaging . Any extra muscle or tissue that is causing the compression needed to be removed as well, if therapy has not helped. And ofcourse the first rib also if that is the culprit.
In my case, the fibrous bands and the anterior and medius scalenes were more the issue than the first rib, as the MR angiogram that they did before closing me up showed a good blood flow in the subclavian artery in all positions of the arm and head. So they didn't remove the first rib on both sides (R) 2002, no issues ever and (L) 2010, some scar tissue issues that are being resolved now.
Every surgeon has their own philosophy, rib removal being the complete surgery but not always necessary, if one can live with some home exercise program to maintain the rib structure and the musculature to give the nerves and vessels enough space to work, but have less complications than rib removal can sometimes be responsible for. But if rib removal is needed, I am not against it, as I had given my surgeon permission if he felt it was necessary to remove it.
Before surgery, my pain level had built upto 9 - 10 all the time and no medications were helping. If the pain level is somewhat bearable and the vascular compression is not so much , then therapy does help some of the vascular component too along with the neural, as TOS is a neurovascular compression condition.