You’re about to receive a powerlifter answer, but I think it’ll help.
Specifically speaking about the bench press in general, your triceps are supposed to help you out significantly. In fact if you hop over to some articles on EliteFTS.com, there’s a handful of articles that specifically deal with sticking points and weaknesses. The triceps along with your lats are extremely important in terms of bringing up weaknesses and strengthening the bench press, and most people come to find out their triceps and lats are trash.
There’s nothing wrong with training chest, but don’t sacrifice your triceps, or developing your lats in hope of thinking a bigger chest equates to a bigger/stronger bench press. Not saying there’s zero carry over, but it probably isn’t going to be as high as you think.
Yes your chest will develop simply by getting stronger in the bench press( take a look at the many top notch benchers), because they do provide aid in the movement. But there’s other ways to make sure you strengthen your chest while still bringing up weak points in your bench, and not sacrificing one body part for another.
I’m not sure what your training looks like either, but if you have an OHP day, you could probably substitute in a chest exercise, or put it in at the end of a bench session for assistance.