Why is proof of waste needed at all? What do you achieve by proving that you have wasted a certain amount of electricity?

Because the chain of transactions with the most accumulated waste is chosen as the "correct" chain. In order to double-spend, you have to cause a different chain to be the "correct" chain, so you'll have to waste even more energy than has been wasted by everybody else, over whatever span of time you're trying to roll back. Thus the more waste, the more secure the chain.

Proof of stake is a newer way of coming to consensus on one correct chain. It took people a while to figure out how to do it securely and efficiently.

Okay, so the chain of transactions that is regarded as correct is the one that has been more expensive to create. Instead of wasting energy, the same result could be achieved by wasting other stuff, such as bitcoins themselves, for instance?

Sure. In fact, that's almost how Ethereum's proof-of-stake works, except it has a protocol that comes to consensus without wasting anything as long as people follow the rules, and only destroys stake when someone provably breaks the rules.

If this is true, PoW is simply a stupid version of PoS and should be abandoned immediately.

If one didn't need to prove they'd done something difficult, then cryptocurrency didn't be a workable idea.

Why do they have to prove that they'd done something difficult? That's what I'm trying to understand.