The error is a result of assuming the coin is exactly 50%, in fact polling uncertainties mean your probability distribution over its 'weighting' is smeared over at least several percentage points. E.g. if your credence from polls/538/prediction markets is smeared uniformly from 49% to 54%, then the chance of the election being decided by a single vote is one divided by 5% of the # of voters.
You can see your assumption is wrong because it predicts that tied elections should be many orders of magnitude more common than they are. There is a symmetric error where people assume that the coin has a weighting away from 50%, so the chances of your vote mattering approach zero. Once you have a reasonable empirical distribution over voting propensities fit to reproduce actual election margins both these errors go away.
See Andrew Gelman's papers on this.
Accepting your calculation at face value implies that if you convince 800 (times some constant independent of n) nonvoters to vote with you, you will... (expect to) always win the election?
Your probability of swinging the election comes from modelling each voter as a fair coin flip, I believe, but this is not really a good model - if each voter is a 51% weighted coin flip then the calculus changes significantly.
Consider my home county:
Expected GDP increase from my voting
... = (fraction of GDP at stake) * P(I swing election) * (P(I'm good) - P(I'm bad))
... = ($400B * 0.1%) * (1/800) * (60% - 40%)
... = $100k
...which seems absurdly large! And it just gets crazier as you look at larger areas, since GDP goes up like nVoters while P(swing) only goes down like 1/√nVoters. For the United States, the same calculation yields a benefit of $300k.
What's going wrong here? (Or, is nothing going wrong? In which case, I guess I'll stop donating to charity and devote that time and energy to Getting Out The Vote instead.)