The problem is that the coin that is provably burned isn't scarce unless the network has reached consensus in the first place, and doing a provable burn of some

The problem is that the coin that is provably burned isn't scarce unless the network has reached consensus in the first place, and doing a provable burn of something non-scarce is non-sensical. Doing a provable burn on a fork of the Bitcoin chain is irrelevant, as nothing is lost, which means it requires already-existing consensus in order to be effective. It's the same with proof-of-stake: it purports to use a chain's own coin -- as opposed to energy -- as the scarce commodity that is consumed in order to reach consensus. But coins on a chain are only scarce if only one chain exists, which means proof-of-stake relies on pre-existing consensus in order to reach consensus. In general, what Bitcoin solves through proof-of-work is actually making a digital token scarce in the first place (by reaching consensus on a single, valid chain). After this, a lot of cool stuff becomes possible, but you can't make use of this cool stuff until consensus has been reached, which means you can't use it to reach consensus.