I agree with going for an escapement mech. However, your comment about not wanting to use a spring seems odd, seeing as springs are what power clocks & wristwatches, not to mention a lot of resettable counting wheels like this in electromechanical commercial pinball machines.
So, maybe if installing an escapement doesn't help, consider an "...
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, attachments, electrical charge, magnetic field, mass, etc., and external forces, spring force, electric charge, magnetic field, gravity, or others.
The term potential energy was introduced by the 19th-century Scottish ...
The block in itself has no energy. If you build a kettle of water around it, the water will not heat up. If you attach axles to it, the axles will not suddenly start spinning.
However, since we spent energy bringing the block up, the law on conservation of energy dictates us that we should be able to get energy back when we bring it down again. So... if we ...
Capable of being but not yet in existence; latent or undeveloped.
The block is stationary and therefore not performing work. If the block never moves, the potential energy remains unused, but "available." Energy was expended in order to place the block at $h$ height. Energy cannot be destroyed, but in this case, it can be stored.
This is not an answer but to question your formulation.
As you can see, due to the flexibility of the rope, at any given time its deformation is undetermined, thus affecting the angle of rotation. There are too many variables, your assumption of linear behavior is not valid. You shall try a rigid steel rod instead of the flexible rope.