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An asic miner should count as one cpu just like my intel core 2 duo.
No, it should be proportional to the work you did. Not trying to hijack the OPs thread and proposal since it probably has some serious merit but no it should not be proportional to the work done. That's the whole point behind one cpu one vote. Some people don't have as powerful equipment as someone else but if they can do one single hash that should put them on equal footing with everyone else. at least that's what the phrase means to me. the idea that we need all the extra hashes from a single cpu is not true, i dont think. just one hash is enough, no more is needed or useful. there is enough cpus out there to secure the network with just one hash each maybe.
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for example, only let each miner perform one hash. whichever miner comes closest to the target is the winner. and they get to mine the block
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You cannot do that. Bigger miners can pretend to be N smaller miners to get more chances. In your proposal, the better connected miners always win, that means of course specialized mining pools with their own servers, not some small solo miner running a single CPU.
They can't pretend simply because it's "one cpu one vote". Unless they go and buy alot of cpus they won't get but one vote. they can knock themself out on getting more cpus though. But I don't think that would scale very well. which is good. 
now of course, you do run up against the issue of how do you make sure that each cpu is only submitting a single hash. i haven't figured that out yet.