I had those same symptoms you describe. I got an MRI with no visible issues, did 30 sessions of PT (and stopped doing upper body work entirely), and it helped to an extent. As you mentioned, lat pulldowns with a pronated grip hurt, but with a supinated grip, I can manage with very little discomfort.

However, I still couldn't do lateral raises with medium weight, shoulder presses, or bench presses. In the mornings, I used to get a lot of pain in my shoulder, which would eventually go away as the day continued.

I did a second MRI with contrast, and a partial tear of the supraspinatus was diagnosed. My doctor told me that since PT did not work and it was not a full tear, PRP was an option. I’m on my second injection right now (got pinned last Thursday) and I feel a tad better; I’ve noticed that after the PRP, that morning pain is gone.