I did the press behind the neck in my teens, 20's, 30's, 40's and a little into my 50's. Now my shoulders hurt like hell. Orthopedic surgeon said I have a slap tear, tendinitis and arthritis in both shoulders. One thing I completely agree with you about. It's flexibility of the shoulder joint. Benching contributes to inflexible shoulders. Then that's where they become painful and prone to tears. The greatest relief from shoulder pain has come from flexibility work and to a lessor extent rotator cuff work. I think for most people benching is the shoulder killer not the press behind the neck that actually contributes to the joints flexibility. Now the only delt press I can do is the military press with a bar.
Shame you got the shoulder problems.
Still you are able to do military press.
Bench press & tight muscles - lack of suppleness / flexibility
And muscular imbalance all contribute to shoulder problems
As well as lack of warming up.