> the weight moves even if there is no 'ground' (say at altitude) in the same predictable fashion"Moves" is frame-dependent. In the weight's own rest frame, it is the Earth that moves.
What is not frame-dependent is the fact that, if there is no ground and the weight (wrong word, as we'll see in a moment) is freely falling, there is no "weight"--the object feels no force and an accelerometer attached to it reads zero. This was the basic insight that started Einstein on the road to curved spacetime and General Relativity: if an object falls freely, it will not feel its own weight.
> how do you model/name/account for that force which causes that to occur?
Using the word "force" here is already wrong as far as General Relativity is concerned: in General Relativity gravity is not a force.
> a rocket doesn't just 'push' either. We can clearly model the chemical and physics behaviors going on there to generate that 'push', and all involve forces which we can account for. None of them meaningfully appear to be gravity.
Yes, exactly. Now apply the same principle to the Earth pushing up on the weight. We can clearly model all of the microscopic behaviors that lead to that push and the forces that account for them--and none of them are gravity.
> gravity also involve forces (spacetime distortions causing very real effects!)
The "spacetime distortions" you refer to, which do indeed cause real effects, are not "forces". That's the whole point. They are spacetime geometry. Spacetime geometry is not a force. That is what General Relativity says.
> what are you even talking about?
I am talking about standard General Relativity, as it has been understood and taught in textbooks for decades now. What are you talking about?