If the pain is strictly unilateral, I would think of Zoster sine herpete (Herpes zoster-Shingles without rash).
WOW, no imaging done. Hmmm...without imaging how would you know exactly WHERE the problem is? You are pretty much guessing without imaging in your case.
Well, try the steroid injection and the physio and see what happens.
Did they check your collar bone? I have torn rotator in shoulder and the pain does travel but not normally down from front & to back. Shoulder pain normally goes up & back. (following nervous system path.)
That is why I ask about collar bone. should ask about it. & some imaging to confirm.
No, i live in UK an dr`s here don`t really go in for doing that with patients. Was offered a steroid injection and physio if things did not get better within 2 weeks.
Have you had any imaging done of your spine or shoulder?
Neurologist referral yet?