Proof of stake to elect a leader that mines using a verifiable delay function seems viable. Proof of stake ensures majority support. And only a single person needs to max out their cpu computing the vfd.
You can rewrite the chain easier than with PoW, but it is plenty hard - requires a majority to precommit to a secret fork. Safe enough for many purposes at a fraction of the cost.
You don't need a secret fork. You just need the community to split over some sort of doctrinal issue, such as already happened with the ETH/ETC split.
Only with PoS, there's no way to judge which fork is the "right" one. In PoW, miners vote with their finite hashpower. There's no practical limit to how many hard forks a PoS system can sustain because they don't rely on anything that's actually scarce.
Because a fork approximately divides the value of the currency between the two parts.
Imagine the USD was suddenly split into two incompatible and free-floating currencies, with the US House of Representatives saying one currency is the official one and the US Senate claiming the other. It would be chaos.
You can rewrite the chain easier than with PoW, but it is plenty hard - requires a majority to precommit to a secret fork. Safe enough for many purposes at a fraction of the cost.