This is the obvious way to do consensus, that doesn't work because of the massive amount of bandwidth needed to have a vote for every bitcoin address in existence. Keep in mind everyone needs to do this checking too. The entire point, and clever part, of bitcoin, is the idea that one can trade bandwidth/digital signatures for CPU-intensive "hash signatures" that are very small and easy to verify.
Not every Bitcoin address in existence. Just those with a large number of bitcoins. And it needn't be for every block.
This is exactly why I say this should be used in conjunction with hashing. A small amount of proof-of-work provides the skeleton and protection against casual double-spends, while proof-of-stake provides an occasional rock-solid confirmation against massive attacks.
Nobody knows exactly how to prevent miners from including low-fee transactions, and hence, making sure people pay fees. Even if we figure it out, if we want transaction fees to remain low enough, there will be little incentive to mine, making the overall hashing capacity low. Thus the network will be vulnerable to attack, and people will not be able to safely send large amounts.
I see this as a major problem, which the inclusion of proof-of-stake, done correctly, can solve.