I actually thought about replying to you in private first, but then decided to write here. I'm glad it didn't back-fire.
I know that you are very well aware about the core/hips movement, rotation, weight transfer etc. And indeed that IS the crucial thing, and the movement of non-hitting (thanks Der, that's better than non-playing

) hand is only secondary, and the "correct" movement of the non-hitting hand only manifests, that the more important thingees (core etc.) have been done/moved correctly. But the non-hitting hand can serve a purpose here, it can get you there. It is because it is so easy to think about it, especially when it is under-utilized. And easy to verify.
So, as mentioned, during backswing, it will help you rotate to the left a bit more. I actually don't think you need to rotate to the left that much more, I think you are rotated just fine, maybe it could be, but only tiny bit more. But it will help to make sure your body rotates to the right as you swing forward, and that the chest is properly "opened" at the time of ball contact. At that time, you can check where your right hand (the non-hitting) is. It should - the hand - be say somewhere near the right shoulder, or heart, heart-shoulder (it is individual, apply rule of thumb). That is a significant distance, from the start position. And if it is there, the chest can't be but opened, the body must have rotated, and then in contact, you transfer the energy nicely, and it all continues a bit (note that if the non-hitting arm didn't do this movement, it can't continue any movement, but it should, it is 1 movement, it is part of it), it all relaxes down (not abruptly), and then the new loop backswing starts...
Even though all this movement, if you look at it, it looks like the arms (hitting, non-hitting) are moved synchronously (both by the rotation of the body, due to weight transfer, as you push with your left foot). And they are. But it should feel as if the non-hitting arm is moved slightly in advance, imagine like if it "charges" the hitting hand by that, and then the hitting arm snaps (like even faster). It is actually synchronous, but it feels a bit asynchronous, like "pre-rotated".
Another thing is, alhought it is not your case. I think often people think about what the hitting arm does, how it moves, what angle, does it wrap around the ball during contact, all those things, which I think is completely unproductive. Thinking about the non-hitting arm, and making sure it does what it should, could free you from focusing on wrong things, and the body - the hitting arm (if left in peace

) - would do the right thing automatically.
About the BH, I don't know. I like it. I would not worry about it. (I try to improve mine too. Perhaps I can't focus on it now.)
Yes, the core/hips movement is crucial. But in your case (only in your case) I'd suggest not thinking about it much for a while now. Try the non-hitting hand now. The pieces of the puzzle will come together...
Understand. When I was younger I also enjoyed more those more mind-less activities

Anyway here I believe, if you start looking at the movement as a whole, there is a lot of fun. And then thinking about movement itself - I mean relative to table - there is a lot of fun and potential too.
Greetings

We all soon gonna play in heaven it seems