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The point is it mitigates the on-chain attack surface which is still prevalent on PoW. Off-chain attacks are still possible for all consensus mechanism.

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What on-chain attack surface? The only way to permanently knock a PoW chain offline is to consistently out-mine everyone else. In PoS, once you lose BFT quorum (1/3 of all votes), it's game over.

> In PoS, once you lose BFT quorum (1/3 of all votes), it's game over.

This is not how the Ethereum PoS chain's LMD GHOST fork choice works. If >1/3 drop offline, you stop finalizing, but the chain keeps growing.

No distributed system is guaranteed to make forward progress if over 1/3 of its voting nodes is faulty, full stop. Once an adaptive adversary controls more than 1/3 of the voting power, they can forever delay the remaining 2/3 of the voting nodes from reaching consensus. Hell, they'd even be able to delay votes from other nodes to slash their stolen stash.

From this, I can conclude at least one of the following:

* LMD GHOST is incorrect

* your understanding of it is incorrect

* LMD GHOST is not a BFT consensus algorithm

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