I don't get the impression people are very interested in actually understanding this as the relevant argument keeps being ignored, but alas - other than the enthusiastic people interested in math would suggest, this actually is the key to understanding the issue

In a vertical cylindrical coordinate system there is a linear axis vertically, the "train", and radial system horizontally, the "flywheel".

For the moment constant altitude change is assumed, so 1g in this direction.

On the flywheel it is classic centripetal ar=Vt²/r. Note, that Vt is meant to be tangential velocity, so the groundspeed.

Then it a simple addition of vectors, into which changes of vertical or tangential speeds can easily be incorporated.

The 'simple addition of vectors' you're invoking here is known as

Galilean Invariance - physics is the same in two inertial systems, and so you can compute in the easy inertial system and then apply a x2 -> x1 + v*t transformation to go in an inertial system that moves with a relative velocity. At schools we train people to attack problems like this, but...

For aerodynamics Galilean invariance cannot be invoked like this.

A system in which you're at rest with the air isn't just as force-free as one in which you're moving with a constant velocity

because air exerts forcesConsider a balloon floating in still air - its buoyancy balances gravity, that's all. Now transform to a system in which the balloon moves downward at Mach 2 through the air- far from being balanced, it will be ripped to shreds because the second system isn't just the same good old force free inertial system.

So we cannot do the simple addition of vectors

because there will be additional forces.

How does that work?

Consider a plane that circles at constant altitude, with the wings an unrealistic 90 degree bank angle. There's some alpha angle, but no beta angle. Superimpose a downward motion at constant speed - suddenly there's a beta angle that lets a side-force appear, and if the plane is sufficiently well-shaped, there's a force driving beta to zero. So you can't simply fly in any attitude that happens because you changed coordinate system - if you change the relative motion through the air, additional forces and moments come up.

So there's no two valid coordinate systems in which the problem can be analyzed, there's only the one in which air is at rest, and the situations 'plane flies a circle at constant altitude' and 'plane flies a descending circle' are different because there are additional aerodynamic forces in the second situation that do not appear in the first.

Now we imagine a situation in which the plane flies straight at 120 kt, or it flies descending at 200 kt (in each case alpha and beta = 0 because of the forces which drive these to zero). Which of the trajectories can be bent into a circle with less g-force (whatever course change you make, you need to add a velocity component 90 deg to your current velocity)?

Standing in a train, facing the loco, you measure the force on the spring balance needed to spin the weight at 100 rpm. By your reasoning the value would be different when the train is moving at 120mph compared to when it was standing at the platform.

Ah, but the train moves the air inside it - which restores Galilean invariance. Try standing on the roof of the moving train in the wind and whirling something around and compare with someone standing at the station...