I'd go even further and say you don't even need to be relatively strong for your weight class, as long as you're not relatively weak. I'm reminded of a good analogy from a John Danaher seminar I went to about 11 years ago. This was right after GSP beat Jake Shields and his main coaches including Danaher, Firas Zahabi and Phil Nurse were there. Someone asked about their views on S&C and particularly with GSP's emphasis on it. Danaher said GSP was a "special athlete" but in general you only need to be about average trained strength and anything above that is nice but will have diminishing returns and come at the opportunity cost of other types of training. He said for example if the average trained fighter can bench 200 lbs (number wasn't important but he wanted to make the point), you could train that lift to be able to bench 300, but the incremental strength wouldn't help that much and your skills and other training would have suffered for it. On the other end, if you can only bench 100 lbs, you're going to be much weaker and need a much higher skill level to compensate - you're better off doing basic S&C to get up to average strength.
Re. bench in particular, I know firsthand that a guy who focuses on a big bench (and nothing else) feels weak as shit when you tie up with him in wrestling compared to a guy with a much weaker bench that does DL and pull ups.
I used to lift heavy on both barbell and dumbbell bench press and IME barbell is more prone to injury if you ego lift. You tend to flare your elbows out which puts a lot of stress on joints - I blew out my AC joint going for 1RM on barbell and couldn't barbell bench without pain for almost 3 years. But I could easily do dumbbell benchpress even with the injured AC joint, because they allow a more natural range of motion. When I felt OK to barbell bench again, I switched to 5-finger "suicide grip" which feels a lot better for me and forces you to keep your elbows in tighter and no shoulder problems since. I now use suicide grip on all barbell lifts including OHP and deadlift.
Re. bench press vs. push ups, if you go light enough to be able to do the same number of reps in each, push ups are superior because they're more of a compound exercise and require you to actively stabilize your core.