I've never understood the fixation on bench press. Why do they even mess with it at the Combine? I think that a deadlift would be a far better measure of total strength or even a squat. In terms of pure athleticism, I think that a snatch would be the best choice - of course, it's too technically complex for the average athlete. And, many would lack the mobility to be able to get deep enough in an overhead squat to catch a heavy snatch.
But, realistically, how often is an athlete going to be lying on their back pushing someone off of them - particularly a football player?
I suppose you could argue MMA when somebody is on their back, but to get a guy off you it's really going to come down to the ability to use the hips to "buck" the guy off. And, I'd contend that pulling strength is more critical to control the arms, wrist, etc. whether you're working from the guard or working in the clinch along the cage.
It's interesting that you never hear anybody bragging about their row - ha ha! And, that's why you tend to see a lot of shoulder issues as people have mentioned here (i.e. bench pressers' shoulder).
By the way, if you want to alleviate the problem, start by checking your technique. You should be retracting and pulling the scapula back on the bench and making the upper back very tight. This will pull the acromion off of the humeral head and give your rotator cuff tendons some breathing room. The biggest challenge in bench pressing is the fact that the scapula are trapped between the bench and the weight of the person. The scapula is important for 60 degrees of motion at the shoulder. This is referred to as scapulohumeral rhythm. Since the scapula is fixed to the bench and it can't move like it normally would if you were standing up, pulling it back & down and retracting it will provide more space for the upper arm to move and help to prevent the rotator cuff tendons and bursa from being pinched under the acromion.
Also, try to keep the elbows tucked into at least a 45 degree angle. And, I would highly encourage people to dump the upright row exercise. That puts your shoulder into an abducted and internally rotated position under load - a recipe for impingement syndrome somewhere down the line
So, how does posture fit into the equation? Well, if you are really rounded forward in the shoulders, then your scapula will be malaligned. Most likely, they will be protracted (abducted) - around towards the side of your body and they will actually be tilted forward - although you wouldn't be able to see this. Again, this closes off that space under the acromion and it also affects your ROM at the shoulder because the scapula will not upwardly rotate properly. I'd contend that in many instances when people have poor shoulder flexibility, it's not a flexibility issue at all. It's related to scapular positioning and that's possibly a function of their upper back posture and mobility.
Anyway, just some food for thought to get your brain going! I'd stick to DB and try to stay in a neutral grip or as close to neutral as possible. Also, I'd do a higher ratio of pulling to pushing exercises. Work on getting those upper back muscles stronger to offset the front side muscles. Couple this with some pec/pec minor stretching and T-spine mobility exercises and I bet you'll see a difference in your posture and your some of your shoulder issues.