The African genes were clearly superior; both theories recognize this fact.
That's not true. Most proponents of multiregionalism that I've read ascribe the dominance of African ancestry to the size of the ancient African population. The issue is that it's not just our genes that are primarily of recent African origin - it's all the rest of the genome as well, and it's not reasonable that all of this was selectively advantageous - including every non-coding region and all the synonymous mutations.
You need significant expansion of people carrying genes to account for this. Genes need people to move them about, and while individual genes with a selective advantage can easily be passed around from population to population without much movement of people, only large migrations can explain whole genomes spreading themselves around.
Alan Templeton's model of human migrations out of Africa, which was based on mtDNA and Y-chromosone DNA, showed the genetic evidence was best explained by three migrations out of Africa - when
Homo erectus first left about 1.8 million years ago, another at 700,000 years ago and the last recent one. Templeton is a multiregionalist, but like everyone else he accepts that we need population expansions from Africa to explain the genetic data. The only concrete dispute between multiregionalism and OOA is the question of how much gene flow was happening in between these major migrations and how big a contribution non-African populations made to the modern gene pool.