I'm a American football player, a lineman but I used to be a kickboxer. What's true for football is more so for fighters.
In football you very rarely get the opportunity to get both hands on someone, brace and win a rep by pushing them off your body. What wins reps is unilateral strength and the base that allows for that. It's basically all about that squat, cleans etc. Most of the upper body is about injury reduction when your applying force at odd angles or as you pass a target.
Fighting is that but more so as speed and power rule the day, not static strength. There's also almost no instance where the press is a variable in a fight.
The bench is a quick way to add some mass and strength in the upper body push, but once the early gains are made I think its way more advantageous to move to unilateral and power based movements.