That might seem an attractive intuition based on observing damage and having an understandable intuition of how damage should relate to energy, its wrong.

Energy is force × distance. For gravitational potential (and, equivalently, the kinetic energy acquired when that is tranformed into kinetic energy by a fall), the force is the force of gravitational attraction between the body and the object it is falling toward (the object’s weight) and the distance is the distance of the (actual or potential, as appropriate) fall.

The nonlinearity you observe in damage is because damage is a complicated outcome that depends on a lot more than aggregate energy involved.

Dropping something from 1 foot has 1/10th the kinetic energy compared to the same thing dropped from 10 feet. Damage is a very poor proxy for energy because there are all kinds of variables and thresholds and structural consideration in determining damage.

A cup might not break at all in a 1 foot fall and might shatter when dropped from 10. The outcome is binary, which isn't useful as a scale of input energy.

10ft vs 20ft is factor of 2 difference in height, 1ft vs 10ft is factor of 10.

Also how it translates to difference in damage depends on how elastic is the collision. If the balls are made of rubber, things get much less dramatic, and energy exchange ratios more obvious.

"Damaging" is not the same thing as "energetic".

I can hit the window many times lightly, with zero damage at the end.

Or I could roll of of those taps into one big energetic swing and break it. There are thresholds below which no damage happens.

lifting something 10 times 1 foot is exactly the same as lifting it 10 feet :)