That’s funny because I do absolutely zero pressing these days. A couple years ago I was using 5/3/1 and I must have put at least 40lbs. on my press but maybe 10 on bench in six months. I did a lot of dips too, I think the heaviest I went was something like 120x5, with a full ROM. Some guys swear by those exercises, what made my bench go up was increasing volume and frequency and doing lots of bench variations like long pauses, floor press, etc. That worked for a while (and I was still doing some pressing too) but I have put maybe 5lbs. on my bench since last summer. Right now I have been looking into some of Josh Bryant’s stuff, he coaches some of the best benchers in the world. He doesn’t seem to be into pressing, and there are other guys with big benches (Matt Wenning for example) who don’t do any overhead work. My impression is that it’s more of an individual thing, it helps some guys but not others.
As far as dips, I would do them except for the fact that I train at home and the dip bars that came with my rack have a crooked weld, one side is about an inch higher than the other. I never really got any carryover to my bench in the past and they were a staple exercise for a long time, but since I haven’t done them lately I would give it a shot if I had a better setup. Maybe I will see if I can wedge something between one bar and the rack to prop it up, unless you have another suggestion.
I’m not a Dr but we can pretend.