So the "finger on ball" case misses the mark on being a good analogy. Thank you for the "early correction".
You're welcome.
If you want to see some truly elegant followups to problems like this, find Marcel Sidi's
Spacecraft Dynamics & Control. It presents a holistic approach in linearized free-body dynamics. Even if you don't understand much or any of it, the takeaway is that if you properly model the problem, solutions you never thought tractable (e.g., fuel slosh) can be tackled. The magic to appreciate in his approach is that the
same formulation can be used to express an astonishing variety of physical phenomena, whereupon the solutions collapse to a surprisingly minimal vocabulary. That's what it was like to be an Apollo engineer.
Admittedly, my base understanding of the concept of "balancing on a center column of thrust" appears wrong. I would expect the physics simulation to confirm this concept.
Learn it the way that seems best for you. Not to put too fine a point on it, but if your simulation doesn't agree, then debug the simulation. This is a solid enough nugget of knowledge in rigid-body dynamics that it can be the yardstick against which to measure program code, not the other way round.
Thank you again.
You're welcome again. And having fulfilled my instructor's duties for today, I will probably not be available for the rest of the day and probably not at all tomorrow.