If you deny potential energy, what other forms of energy do you deny?
Here's a list: https://en.wikipedia.org/wiki/Energy
Does a 10kg bowling ball at 10m above the ground "have" potential energy? Or does it have the potential to gain that energy if you let it fall 10m?
It depends on the definition of 'h' in the gravitational potential energy formula of U=mgh. Height isn't an absolute thing, it is measured between two points.
If I move that bowling ball slightly to the right, where there might be a 10m deep well... has it suddenly gained twice as much potential energy just from me moving it less than a meter? In the formula 'm' and 'g' stays the same, but 'h' has changed because the reference point for the measurement has changed.
Things like how many kJ are in your sandwich (chemical potential energy), or how many megatons of TNT are in your warhead (nuclear potential energy) don't change just because you move your lunchbox by 1m or launch your missile. Some measures like kinetic, electrical and gravitational potential energy depend on what chosen reference for "zero energy" is.
If I am on a train travelling at 100km/h and drop a bowling ball on my foot, does it hurt more than when the train is stopped, even though when the train is in motion the ball is travelling at over 27m/s?
I guess you could chose to use the earth's surface as reference point for kinetic and electrical energy (making the assumption that the Earth is electrically neutral and stationary), and you could use the center of the Earth as your reference point to give a more 'absolute' measure of gravitational potential energy.
That may work for you, unless perhaps you are in the business of launching bowling balls into the sun, in which case they have far more kinetic and potential energy than those calculation would suggest...
EDIT: on second thoughts, maybe you can establish of what electrically "neutral" is using electrostatic attraction/repulsion, I guess, but would be pretty hard to achieve a useful reference for single digit voltages...