I'm quite lost as to the whole point of the scheme. Blockchain consensus doesn't need to be faster and there isn't any issue with it right now.
Work is applied to a block hash so that it's highly unlikely to produce a valid hash if it's modified. But this forces work to be done after the block is created, resulting in painfully long delays. The goal is to do the work before a block is created, without leaving the block subject to modification.
The "delay" that you're saying presumably refers to the block interval. That is by design and natural variance will alter the interval slightly and on average blocks are still generated with a 10 minute interval. If you want faster blocks, modify the difficulty mechanism to produce a shorter block interval.
A block's list validates the pubkey of the following block, and everything in the header is included in the signature, which only the miner can produce. Would this not secure the block as well as a block hash with POW?
The whole idea of POW is that miners are hashing everything in the block and are in essence exchanging the work done to mine a specific set of transactions. This ensures that the whole game theoric approach still functions and including transactions after the block hash is already found will violate this logic. This makes block reorganization trivial from an adversary perspective, having the keys also means that you can replace the transactions with zero difficulty as and when you'd like. Building an alternative chain requires no work since you're not hashing the transactions, but finding the hash of the public key.