I think the whole idea of "punching through" your target can lead people into some bad habits. I've seen a lot of people defending pushing punches by saying they are just trying to "punch through the target." The theory is correct; you want to follow through on your punches, but you don't want to overdo it, and that's what many people get in the habit of doing when they overthink these cues.
I've heard trainers equate the force transfer of a punch to a car crash. Shout out to Barry Robinson here, who provides this handy metaphor. The damage of the car crash is done on impact, but often the momentum of the crash will cause the cars to keep on sliding and grinding together. The car crashes, then "pushes" the car further, but no extra damage is done there.
Similarly, the damage of a punch is done on impact. The "push" afterwards that people often end up doing doesn't add anything of note. Of course, the impact needs to be "deep," but that is accomplished by just trying to punch hard and having your mechanics down. I don't think you're gaining anything from lingering and actively trying to "push" anything.
Barry also uses the cue that states "you press a button" with a body punch and then go right back to work/guard/control/whatever. I'm not going to waste time "pushing" into a guy's body with my glove when I could be using that hand for defense, control, hand play, or doubling it up. The push is just tiring and doesn't produce any additional force. It might be uncomfortable to have someone "push" into your ribs/sternum/whatever, but whatever. It's "uncomfortable" to jab a guy and then prod his face a bit, but it's not going to cause extra damage, and unless you're actively controlling with that touch you're just creating a defensive liability.
I need to be careful here, because penetration has to happen. Pushing is the problem. One of my favorite body punches is a stabbing right uppercut if I can slip outside the jab. When I throw it, I think of violently running the guy through and then immediately retracting the shot, but I don't think "pushing" is the proper descriptor. Big, explosive difference between pushing a knife into someone and stabbing them. The penetration happens too fast to be called a push, which has more of a grinding/slow connotation to me.
