It feels that you've got some idea that I'm not understanding the difference between energy and momentum because it's a common misconception, and hold on it. I do understand that they are different. My objection is your insistence on them being "unrelated".
As of the yo-yo, let's remove the muscle power and the wobbling earth out of the picture and consider a it a closed system.
We've got a fully wound-up yo-yo, not rotating. It has a certain amount of potential energy. When you release it, its potential energy starts transforming into kinetic energy of linear motion and of rotation. This kinetic energy can be measured at any moment via observing the linear and rotational momenta of the yo-yo, which are the functions of its mass and torque, and the both velocities. Speaking of which, there is no other way of measuring the energy of this system. As it reaches the end of the line, and starts winding up again, its potential energy is zero, its kinetic energy is at its maximum, and its linear momentum changes the direction to upwards.
I'm telling you this to demonstrate that I understand the difference and your main objection is not exactly applicable here.
And, of course, momentum and kinetic and potential energy are intimately related in such a system. I don't understand how one could deny that.
> Momentum has nothing to do with energy
That's what I meant. Kinetic energy is a function of momentum and you are insisting it is not!
Perhaps we are being confused by each other's different ways of using the word Energy. When I use it (in the mechanical context), I mean strictly kinetic energy or potential energy, but nothing else. I've been taught to use it that way and was quite harshly slapped on the wrist (verbally) for failing to stick to it (that is, for magical thinking).
You seem to be using it in a broader sense (e.g. "nuclear energy". I don't know what nuclear energy is -- it s what multiplied by what, specifically?).