They don’t need to be, but it gives the best security guarantees. PoS is subject to threats that don’t exist for PoW, e.g. https://news.ycombinator.com/item?id=27672759.
That is not true, the security has nothing to do with energy consumption, only with cost. If you had to verifiable set a million dollars on fire for mining a block, that would be as secure as proof of work without wasting energy. Also anything like mining is only necessary at all if you want anonymity, if you drop that requirement you unlock a vast space of very cheap solutions.
Sure, but nobody has found a way to do that, nor has anybody found an alternative to PoW that doesn't have terrible security tradeoffs. It doesn't help to say something magical could have better properties - you need to find something that actually works in practice by satisfying all requirements
https://eprint.iacr.org/2021/872
This doesn't solve the issue entirely but it does provide a mechanism for recovering in the case of a major attack.