This bugged me so much I was forced to abandoned all my exercise therapy knowledge and everything I had ever read about training and had to turn to my physiotherapy routes.
I first went back to basics I looked at the movement patterns of my pressing technique. Unlike big bench pressers like Ervin Katona and Derek Poundstone where they press in flat shoes, have short limb in relation to myself (This is not a sizest thing as I have great respect for these guys. It allows them to create bigger forces as they have smaller levers.), and get a massive amount of drive through their chest and triceps. I press in a more upright position I get my press from the power of my jerk and the fast transition from my deltoids to my triceps. I have always pressed in the same shoes I squat in, so I get the power from my legs and also because when pressing in flats I tend to loose my balance. I also have a narrow stance on my press which gives me less stability when trying to utilise my pecs and forced me to use my triceps as I can't extend through my lumbar as much and maintain my stability.
So for me bench pressing wasn't ideal to help improve my shoulder pressing technique so to improve I had to improve the strength in the transition phase between the delta and the tricep phase of the press and to increase my tricep power.
So to increase my strength in the transition phase I do lots of static seated press and either floor press or close grip bench to increase my tricep power. Skull crushers are also great but ideally you need 2 people and I train on my own so it's a risky exercise if you want to push harder. Also a big tip for overhead pressers is to make sure that you have strong upper trapezius as this will help in stabilising your shoulder girdle.
I'm finding as I'm getting stronger that I'm starting to utilise my chest more on pressing and I'm starting to do more upper pec work and have a wide stance when pressing. Whether or not this is a natural progression or me developing my chest to press over head I'm not sure. Probably a combination of both.
So overall based on my knowledge about anatomy and human movement I wouldn't suggest that anyone stop bench pressing but if your a keen bencher and you find your struggling with your press there are a lot of variables you need to look at, I.e. shoes, stance, position you press in, your body shape, etc