Anytime you have a vote with more than two alternatives you run into problems. Any form of approval or average voting is just awful, awful, Awful! Condorcet voting is just to complicated. So here is my suggestion:
Split the voting up in multiple yes/no votings.I have been in a national's party's congress voting for a new logo and that is how we made it; Make a cup out of it. Logo A vs. B, C vs. D, E vs. F, G vs. H. (Vinner of A vs. B) vs (vinner of C vs. D), (Vinner of E vs F) vs. (Vinner of G vs. H)... you get the point. In sports that system is ineffective because placement beyond the first depens on which pairing the team gets. But for this voting it is pefect because we only care about ONE alternative - the vinner.
One disadvantage is that it takes some more time than other voting methods and that is why it is not used for large scale votings. It requires 13 separate votings for 16 alternatives. Another is that it requires some attention from the voters. A third that someone might object to how the alternatives is paired against each other. Should X face Y and Z face O, or should O face X and Z face Y? That problem is solved by letting a computer randomize the pairings.
But we can run the voting rounds simultaneously. If we have 16 candidates we can first run the 1/8's voting rounds simultaneously. Then the quarter final round, semi final round and the final round. That is four rounds and if we let each round take one month we are done in January. Yes, that is a long time but a decision that involves so many people and seems to be so important should be allowed to take the time it needs! But we could instead have each round being two weeks long and be done in eight weeks which definitely is not to much time.
We could even have used that method for the first voting stage.
Regardless of which method we use the last voting absolutely, positively, definitely needs to be "Should this logo replace the existing logo?". Maybe that question should have been the first to vote about but it's to late to change that now.
BL