I also have bilateral thoracic outlet syndrome.
Two causes (usually)
1. cervical rib
-remove said rib
2. tight scalenes, pecs
-stretch said muscles
massage with pin and strech and trigger point therapy on key muscles is key in solving the shoulder problem.
stregnthening isn't the key in my book. re-wiring your shoulder's firing patter is. get those scapula stabilized, upper traps inhibited, lower traps activated, serratus anterior activated, pecs, subscap, stretched....blah blah blah. Most climbers I seriously doubt have any weak shoulder muscles. over active ones though. and as for labrum...you can sometimes calm the shoulder down enough to keep the humerus from grinding it, but it's a common surgery. impingment is more easily solved (see above) as is frozen shoulder. Supraspinatus and biceps tears are manageable, unles a big tear, then once again surgery.
come see me when I get my license. I'll help y'all get better. I'm specializing in shoulders, knees, feet, wrist/hands, and elbows....besides the spine.