It's really a fantastic technique when done properly and trained. Just like any technique it's gotta' be trained. We trained it a lot.A shove is not usually thought of as an actual MA technique - because it usually isn't. But, I suppose it can be when executed as you describe. There are a couple of kata containing a double palm heel to the lower and upper torso after an initial circular clearing move. These are designed as pressure point strikes, but with more follow thru and some slight forward motion, can easily be called a MA shove. What differentiates a MA shove from the common shove, is that the body stays centered over the feet, without leaning into it, and originates from the core, as you touched on, and puts you in position to take advantage of the change in engagement distance (maii) to follow up.
Bob Campbell, a ninth dan in Uechi, and one of the first people I ever saw doing Martial Arts, has been living in Hong Kong for quite some time now. But he would visit back east every year or every other year. He'd always have a seminar for Uechi guys when he came. A bunch of us always went even though we weren't Uechi, but we knew a lot of those guys. We always had a blast.
About six years ago Bob showed us a technique that I had never seen - but it was the first cousin to a shove and I picked it up pretty easy. I'll see if I can describe it correctly.
You're in a right leg back stance, half way between a front stance and a side stance. Picture a right hand reverse punch, vertical fist, to mid chest or sternum.
But just as it starts - your left hand grabs your right wrist/fist (thumb beside the forefinger, not wrapped around the wrist) and with full body behind it, and with whatever step or shuffle you like, you use it like a battering ram, both arms propelling that fist with your whole body behind it.
Man, is it great. By the next day down the dojo I was having guys holding shields in front of their body and knocking them down. And not shields with inner tubes, foam shields.
To me, it's the same core principle of a two handed shove, but with a smaller, and harder, impact point. It really moves someone. And it fricken hurts, big time it hurts.
And, yeah, I know I always preach "hands up at all times". But this can be used from a relaxed stance with your hands already in position one over the other, right in front of your groin, forearms lightly on your hip points.
I hope I described it clearly. It's not as complicated as I might have made it sound.