I agree that for an impingement you should suffer from other neurological symptoms. And even then it isn't said that it is the reason, because mindbody symptoms can be very diverse.
You see a lot of people suffering from TMS (RSI in my opinion) who have pain in their neck, shoulders, arms and hands, the whole nerve path that originates from the neck seems sensitized and the muscles in play feel sore and stiff. Often tingling or 'electricity' is felt in certain area's (fingers, wrist,......), or numbness and difficulty bending/stretching certain fingers is experienced. It seems to come up during certain work where those muscles have to act, like using the mouse behind a computer or doing other physical activities.
The fact that you have to do lots of overwork could point to TMS imho. Personally I hate bosses who let their staff do overwork on a regular base, maybe you do too?? Or maybe you hate yourself for not saying "NO, not tonight!" to your boss. Maybe I am projecting my own experiences on you, sorry, but maybe it hits home

I discovered that I did my work in a strained way. I used to do work at night regularly, felt people breathing down my neck because they of course always needed the documents yesterday, felt irritated when more crap was dumped onto my desk without proper consultation, felt irritated about dumb, time-consuming projects or work that was the result of lazy co-workers not doing theirs properly. Not speaking out to my boss and setting clear restrictions on what I could/would do as an employee, was one of the major contributors to my TMS.
One major give-away that it is really TMS: it is experienced in both arms, either simultaneously or at different moments in time. If so, impingement is very unlikely. I have talked to more than one person who had an operation to get rid off the symptoms, only to get the same symptoms in their other arm after a couple of months. Chances of that happening because of a real, structural problem are slim to none.
I think I understand what you are proposing. Can a real, structurally induced pain give rise to TMS, because it enrages and scares you? Yes, I think it can, but I highly doubt it is what you are suffering from. With that I mean that I think your symptoms are not the result of something structurally wrong in your arm. That said, I am not a doctor so you have to decide yourself which road to travel.
good luck to you, hang in there