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If you checkout the Horde Chess rules section you will see that there has only been 12 draws from nearly 500 games, about 2.5%, whereas black has won about 25% and white the rest. This would not have a dramatic effect on the balance.
I guess I am a bit of a bunny as I have played 17 games with only 3 as white, but I (he says puffing out his chest) won 8 and only lost 6 of the black games, a little better than the average stats.
Perhaps another option is that this variant is only ever played in pairs of games. Maharajah Chess is even more one sided as black can force mate from the first move.