If someone has a specific injury such as a strain, I would NOT advise SMR/stretching on that SPECIFIC region.
Otherwise, some type of self myofascial release and flexibility/mobility work is something most people should do. I'm not going to recommend any specific programs out there because I see flaws and strengths in pretty much all the popular ones. So this is where you will have to do the research and see which protocol is the best fit for you. The people I train have the advantage in that they can give me immediate feedback on a certain move and we can make subtle modifications as needed. This type of subtlety is logistically improbable via an online forum.
Just know that:
1) Not unlike lifting weights, there actually is a right and wrong way to go about it. For example, when foam rolling the thoracic spine, it's dangerous and almost always unnecessary to move down to the lumbar spine.
2) Do NOT expect immediate results. Remember - you're trying to correct years of improper alignment.
By driving the back of the head into the bench, you are engaging the cervical extensors; this should be able to somewhat disengage the cervical flexors, one of which is the sternocleidomastoid (it has other functions in addition to this, btw). So it might be something worth experimenting with. Just know that its just one piece of the puzzle and not a magic bullet. Same with the arch.
My post yesterday was based on your description that inclines and military pressing does not cause pain. That's why I proposed the theory that downward movement of the clavicle is a part of this. Which is why I proposed looking into the sternocleidomastoid as well as the subclavius.
The good therapists will have various methods of addressing trigger points (and I mentioned scar tissue as a worst-case scenario). Now some of these methods, IMO, are about as helpful as shaking a bag of chicken bones and uttering gibberish at the stars. However, to categorically state that ALL modalities are worthless is both absurd and irresponsible. So you're going to have to take it upon yourself to interviewing therapists in hopes of finding a competent one.
It bears mentioning that, although it could be the sterno, you will need to address how you stand, sit, move, and lift. Something happened through the years which culminated in this issue.
So take a comprehensive approach and do not be one of the ignorant fools who thinks all anyone needs is a barbell. Others can debate this subject while people like me and my clients are training without pain.
When performing curls, what does the weight do...? It pulls the clavicle downward. The clavicle also moves slightly downward at the finish position of the chin. Now if the pain, when doing chins, is not at the finish position, that could suggest irritation at the brachial plexus. So, your hypothesis of this being related to thoracic outlet syndrome may not be paranoia.
It's your body and you can't trade it in for a newer, shinier model. Now that you're better informed, get tested, ask the appropriate questions, and get back to us with the results.