Have you tried benching with a neutral grip, for example with a football bar or dumbbells? As others have already stated, only powerlifters and football players prepping for the combine absolutely need to bench. For everyone else, it's a very good exercise if you can do it without injuring yourself.
Another option is something like the EliteFTS shoulder saver pad. This fits over the barbell and limits the range of motion, effectively turning it into a version of the board press. If the bottom part of the bench press is causing you problems, this might be something you can try out.
Yeah funny you say this, because I've been floor pressing with a trap-bar (neutral grip obviously) and it's generally felt better than benching I'd say...but I still have a bit of a tinge/pain on my left shoulder.
But I also hurt both my shoulders and particularly that one pretty bad when I started doing BJJ again, so it's not because of the lift itself...I probably have a partially torn rotator cuff or an impingement or something stupid. Not a big deal though.
What I'd add to this topic for OP, to add one to some others, is yes Bench press is NOT necessary at all. If you're a powerlifter? Yes it's a competition lift. If you play HS/College/Pro football? Yes you're going to be doing it in the weight-room unless you're a stud who gets an exception maybe or claim bad shoulder issues, maybe.
@Jkillah Judging by your avatar I assume you do Judo or BJJ correct? So I will assume that is the goal here, to strength train for grappling specifically or to look good or for general strength. In which case - ditch the bench press for sure. Specificity is kind of a "dumb" thing in the lifting world, but I do actually believe in this case that floor pressing >>> bench pressing for the purposes of grappling/BJJ, probably for Judo as well.
Weighted chin-ups / pull-ups are basically king 1A/1B here too, imo. So if benching hurts you, I'd move to neutral grip floor press (trap/hex bar), or neutral grip DB pressing, or simply do overhead press with pronated/neutral grip + the pull ups (weighted if possible). Dips are great as well but with poor technique will probably shred your shoulders.
Someone made a comment about push-ups which was kind of dumb, but actually good at the same time. Because at least push ups will rehab/prehab your shoulders by working on stabilization and shit, whatever the exact scapular/rotator cuff details are.
Probably what you are missing is band work as well. Rotator cuff work - external shoulder work, posterior delt work with face pulls and shit...that is probably the root cause of your shoulder pain as it often is, the rotator cuff and weakness of it and the rear delts.