Disc issues are rarer than you might think. Disc issues shout that something is wrong; to date your symptoms are a whisper, correct? The vertebral column is very well designed. It permits trunk extension, flexion, rotation. The vast majority of nerve problems occur outside the protection of the vertebral column. By necessity nerves must intermingle with muscles and their fascial coverings.

Your symptoms so far are light, and may never get worse. The mild compression of the brachial plexus you are suffering from is most likely caused by trigger points in scalenes, pec minor--anywhere along the path from C5/ T1 to the tips of your fingers. Before even considering other more drastic treatments you might visit a massage therapist trained in NMT and MFR. Excessive stretching may exacerbate irritation of the nerve sheaths.

TOS may not be the cause; it could be irritation in the carpal tunnel.

I think we've all at one time or another experienced mild nerve compression; whatever the cause, the body usually heals itself if not interfered with too much. Keep in mind that if we push our doctors, for your peace of mind they might prescribe against their own best judgment. IOW, trust your doctor till he gives you reason not to.