Its not a good idea. It would make Bitcoins follow an unfair distribution. By proving you have stake, which means, many bitcoins, you would simply be proving that you're rich and then the system would be only rewarding the already wealthy to validate transactions, and there would be little incentive to add computer power to the network, which is what the 50 btc block awards are designed for, to be a reward and a stimulus to add robustness and block validation to the network. It would create a rich-get-richer scenario, and bitcoin distribution is already following a power law.
This alternative wouldn't need computing power - that's the point.
Furthermore, if you want more influence in the current system, all you have to do is buy yourself more computing resources. Morally not much different than buying yourself more bitcoins AFAICT.
Lastly, if the bitcoin-rich (as opposed to the computing power-rich) started "voting" themselves more new bitcoins than the allowed amount, then people would lose confidence in the currency, and the bitcoin-rich would no longer be rich in any meaningful sense. They have more incentive to maintain the currency's value than the current miners do, it seems.