you could, theoretically. You cold also create man made natural lakes in places where it's convenient, but again, you need the rough shape of a lake high up, and the rough shape of a lake near enough, and lower to the other lake, that you can generate significant amounts of power.

You might be able to use an old quarry, most old quarries are flooded anyway, but idk of many quarries that exist near the base of a mountain with the ability to have a higher elevation storage pond. Most quarries are probably going to be pretty close to ground on account of being a quarry, so that doesn't really help much. You could also use a mine, but the problem there is going to be getting water out of it trivially. It's just not really conducive to that use case.

Another big problem is going to be pollution to the water supply, especially in something like a quarry or mine. That's a non improbable issue.

Ok so TL;DR here, is that you need an incredibly rare formation of geological features, and the ability to use them. If you've ever played minecraft you can think of it like finding a 3x3 formation of bedrock on the top layer of bedrock in the overworld. If not you can think of it like staring at a TV displaying static waiting for it to suddenly emerge an image that isn't just indecipherable noise. Or better yet, the monkey on a type writer example, given enough time, it will produce the entire works of shakespeare.

It's theoretically possible, but practically, when possible, it's probably already been done, and if it hasn't there's probably a good reason for it.

If you're analyzing this from a mathematical perspective, the problem here is not finding two lakes, or finding two lakes at different elevation levels, it's finding two lakes, at different elevation levels, sufficiently close enough to be practically uesful for pumped hydro.

The reason why we're using batteries right now, as i've said is partially two fold, it's a lot simpler, you can invest the majority of your capital into an energy storage medium, the remainder is for infrastructure, land, buildings, power equipment, which makes it a very low risk investment. The second primary reason is simply accessibility. Chemical batteries are simply the most prevalent, cheapest, and most accessible form of energy storage right now. They work anywhere, they can be built to any scale, they work no matter what time of day it is, or what time of year it is. They literally just work.

If you're trying to do something like pumped hydro you're going to need a massive, multi billion, possibly hundreds of billions of dollars of capital, probably close to about a decade of construction, and only then after the construction can you start it up and start to generate power.

With a battery storage system, once it's built and approved for use, it can immediately start providing power storage. And for the cost of a few million, to a few hundred million. And again it scales basically infinitely.

There are also a few other problems, Digging a big hole and filling it with water while it sounds simple is more challenging than it would seem. You need a geologically suitable area for it, and at this point you're probably going to be flooding a dammed mountainous area anyway, so it's probably redundant. There are earth fill embankment dams, however they have issues with subsidence for example, and that tends to be rather spooky. You would experience a similar issue here, the only difference is that it's below ground, instead of above. What do you do with the removed material? What do you do if you're in an area with a lot of bedrock? You've basically just created a quarry now.