The dialog can become somewhat convoluted, perhaps I am the blind person that cannot see what you see.
Several points I agree with, but some of your points I do not grasp fully.
The ones I agree with are:

a) “ A mass when moving has kinetic energy.
b) “If you carry a mass to the top of a building, it is said “to have potential energy mgh relative to the ground.”
c) “if that stone drops from the building it will hit the ground at a certain speed, hence a certain kinetic energy 1/2mv**2. That is real energy, capable of doing work,
d) “Potential energy is not a thing”

The following points that you make are unclear to me, "if" I try to understand them from your perspective.

a) “When a mass moves, it acquires kinetic energy.“

I would qualify this by saying that; when a mass moves it “contains” kinetic energy and when it “accelerates” it acquires kinetic energy.

b) “The potential energy is only a number that tells you how much kinetic energy you’ll get after the stone drops. It has no reality in itself.”

I agree that the numbers will tell you the amount of kinetic energy at any point in the acceleration path, but the numbers are not the cause of the acceleration, the cause (like you said ) is a “force” and that force exists in a “field“, such as a gravitational field. Since the field contains the force, and causes the acceleration, the field must contain the “potential” for the kinetic form of energy. This explanation does not negate the use of numbers to express quantity, It simply states that; a force within a field is needed to accelerate mass. In other words the forcefield contains the “potential” for (kinetic) energy. In the case of a falling object, the forcefield is gravity and in the rocket propulsion it is usually caused by the forcefield within the chemical structures, although in some cases by electronic and/or magnetic fields.

So: "Que la fuerza esté con vosotros", como dicen en Star Wars.

This type of a friendly dialog about differing opinions is what I enjoy about the HZ.