But when Henry Rowland spun a charged ring then he was able to detect a magnetic field with a compass ( here is an AI slop description of this experiment for truck drivers ).
Later, Wilhelm Roentgen and Alfred Eichenwald refined this experiment using different setups with charged spheres or cylinders.
They also detected a magnetic field caused by the motion of electric charges on the surfaces of these spinning objects.
Are you claiming that spinning charged objects do generate a magnetic field but this field is somehow different so that when it is made to vary (by varying the angular velocity of the electrically charged object that generates it) then this time-variant field is unable to induce EMF or current in a detector coil ?
... or do you disagree with Rowland, et al, that spinning charged objects generate magnetic field at all ?
Read what I wrote more carefully:
"I've just concluded that a rotating charged ring,
which we know creates a magnetic field because it's a rotating current, cannot produce induction in a coupled circuit when its rotation oscillates around the axis,
even though there's a variable magnetic field (B) and a change in flux"
I agree with you that a charged ring does indeed produce a magnetic field; this has been verified experimentally.
However, if its angular velocity varies, it will not create induction despite the fact that the magnetic field is variable. This implies limiting the validity of Maxwell-Faraday's equation to currents from a generator. Only currents with spatial variation along the circuit can induce a current in a coupled circuit, so not the current from a charged ring. This is my thesis.
A snapshot of a charged ring rotating at variable speed shows no variation in current along the ring, while a snapshot of a circuit powered by generator current does show variation, due to the finite propagation time of the signal in the circuit.
The only way to invalidate this idea is to produce an experiment demonstrating a current induced by a charged ring at variable angular velocity, not to invoke the theory that the idea challenges.
This experiment is difficult to carry out. It is much more interesting to use the idea to produce effects not expected by Maxwell-Faraday's equation, which will validate or invalidate the idea depending on the results. That is what I am working on, and one of the effects is the possibility of inducing a direct current. I have the mathematics that allows us to move from a propagating current gradient (the case of an ordinary current in a circuit, not that of the charged ring) to a charge density gradient, to its electrical dipole field, and to the influence of this field on the induced charges.
As for the magnetic carousel, I answered the question with my method. If one thing is legitimate, it is to grant everyone the freedom to use their own method to solve a problem. The choice of the reference frame in which the analysis is carried out is part of this.
If it is not the solution to the problem that interests you, but a difficulty or a possible paradox in a particular framework of the analysis, you will need to rephrase the question.
One final remark: the subject of Archimedes' screw that I raised earlier, as we will have understood, concerns this missing link in the induction of a direct current. The posts I have just made here, following a recent insight triggered by the question of the charged ring, even if they do not respond to other posts that may have been made recently, are along these lines and are perfectly relevant to the subject.