So, the top X scoring stories move forward, regardless of what heat they are in?
I think that will be less fair, for several reasons. The first is that the number of people voting in each poll may vary. So what do we go by? Absolute numbers? But that will penalize heats voted in by fewer people. Percentage of total votes? But that will give more value to individual votes in heats voted in by fewer people, and may skew the results far more than the current system.
Also, beyond total numbers, there's no way of knowing that the same people vote for all the heats. If people join the contest late, or drop off, they might only vote for some heats (or some people may only vote in the heat their own story is in, etc.). And since different people may have different scoring habits (and even the same person may shift their criteria over time), we shift the issue from being one of which stories are in the heat, to one of which people voted for which stories. Which is inherently less fair. After all, the winners of the contest should be the best stories, which means that in theory they should be able to beat *any* story. True, the heat system means that a weaker story may make it further into the competition than it otherwise would, but that's self-correcting. Losing a story because its heat was voted in by less generous voters, however, it entirely arbitrary.