[
having ATOS almost always means a rib resection? ]
My thoughts on it..

There are differing amounts of severity for all types of TOS.
Some will have mild, moderate or severe symptoms, and sometimes it is not cut and dried as to which type a person may have.
Is, or was the therapist that you had one with advanced training or specialized with TOS type syndromes?
PTs w/advanced training and experience can be very helpful in locating what parts of the body might be causing problems.
And don't forget trigger points & myofascial adhesions can cause many pesky problems that can feel quite severe, but with proper tx can be resolved.
Does your neck/shoulders seem to be more tight than usual when the cold feeling happens?
Finding a expert/advanced PT, body worker, or even expert DC that does PT type modalities is my suggestion.