The gyroscope and its ability to avoid "falling down"

[Dale]

So we are forced to say that acceleration is not proportional to force in general but only to the "net" force

Yes, that should be clear to anyone who has studied Newton's 2nd law. The f in f=ma is the net force.

So not all forces are proportional to acceleration but only a certain type of force.

The net force is not a type of force. It is the sum of all forces of any type acting on a given object.

Imagine that the stone is not there and that the rope is formed by a series of elastic rings hooked to each other, in a row.

Then we grab one end of this sling with the hand and let it rotate: the rings lose their roundness and lengthen in the radial direction, on one side (the centripetal one) and on the other (the centrifugal one).

Here is the unequivocal "measure" of the two forces, the rings that lengthen!

It is an extraordinarily clear proof: the rings are longer (on one side and the other) because there is a force that pulls on one side and another force that pulls on the other!

Note that the two forces you describe here act on the rope (or the rings), not the stone. This does not contradict anything that any of us said above regarding the forces on the stone, and I suspect that nobody would object to it.

I do not intend to suppress my ideas to adapt myself to the opinion of the majority

Stubbornly clinging to misconceptions is not a virtue, particularly when correct concepts have been taught to you. You are choosing ignorance instead of knowledge. It is not a teacher's job to exhaust themselves trying to shove knowledge into an unwilling recipient, and a student who demands that a teacher do that in order to teach them is simply being selfish and inconsiderate of other's efforts. You, in particular, have been taught correct principles here, but for whatever reason you are unwilling to learn them. That is fine, it is not mandatory for you to learn, but participation here is a privilege that this community reserves for those who are willing to learn from the community.

For future readers who may be interested, the math that I had requested @Luigi Fortunati to work out is fairly straightforward. It is easiest to work in polar coordinates, the acceleration is given by equation 4 at https://ocw.mit.edu/courses/aeronau...fall-2009/lecture-notes/MIT16_07F09_Lec05.pdf

$\mathbf{a}=(\ddot{r}-r \dot{\theta}^2)\mathbf{e_r}+(r\ddot{\theta}+2\dot{r}\dot{\theta})\mathbf{e_{\theta}}$

For simplicity, if we let $\theta=\omega t$ and $r=r_0+b t$ then $\mathbf{a}=-r\omega^2 \mathbf{e_r} + 2 b \omega \mathbf{e_{\theta}}$. Since $-r\omega^2$ is always negative, the radial component of the acceleration is always directed inwards.