If you picture a dense weight like a cannon ball on the end of a string you're right, but if you're digging down n meters, encasing n/2 meters worth of dirt and moving it up and down the free n/2 meters of shaft, the energy storage would indeed be proportional to n^2.
I don't know anything about the field and had the same reaction you did, but considering parent is running a startup in it they're either a lunatic that doesn't know the equivalent of FizzBuzz or there's something we missed on first inspection, and we should charitably assume the latter...
The single shaft vs multiple parallel approach does seem a bit risky in the early days. If there's a 10% failure rate, and you built one shaft, that's a 10% chance of an existential threat to the company. 10 shorter shafts mean one will likely be inoperable.
Of course over the long term worrying about this doesn't make sense. Once you've scaled, 1k large vs 10k small shafts would not matter from this perspective.
Best of luck mate!
There's another thread on this comment page talking about why height is important. Please see this slide in our presentation illustrating it. https://docs.google.com/presentation/d/17FI-jrI9RWS3q7Ng44Yh...
Gravitational potential energy is (approximately) linear in height. I say approximately because this assumes constant g (which is a good assumption when h is small compared to the radius of the earth, which it is).
And in fact, a consequence of gravity's 1/r^2 nature is that one is only subject to gravitational acceleration from what is beneath them (shells above cancel out), so mine shafts are less efficient than towers (the effect size is small to the depths we can mine).
So adding more height doesn't help, and if that height is underground it could actually hurt net efficiency.
Efficiency in this context refers potential energy stored per unit height. The field is conservative no matter what you build.