I feel like this is a separate issue. If you don't feel "chest" exercises in the chest, you need to make better exercise choices and/or use specific techniques. That could mean pre-exhausting the pecs, focusing on dumbbells instead of barbells, playing with angled work, etc.
Skipping chest work just because you don't feel the chest working is throwing the baby out with the bathwater. That said, plenty of guys got big and strong without a ton or chest work. Old school physical culturists mainly used floor press and push-up variations and flyes as their only "chest work".
Plenty of Olympic lifters used to only train the two Olympic lifts (overhead work) and their direct variations. I'm also pretty sure standard issue CrossFit has minimal "chest work" outside of a shit-ton of dips. Those guys/gals get in solid shape.
Dips and incline presses are sometimes used as assistance for building overhead press strength. So even if wanted to prioritize shoulder pressing, you may benefits from some strategic "chest work".
Going from one extreme (lots of chest work, minimal back work) to the other extreme (no chest work, plenty of back work) can be just as bad for posture. Harder to do, but possible. Muscle imbalance is muscle imbalance.