So are females most strongly attracted to the tough, macho, white-striped males? Actually, no. Lab studies have found that females of either morph prefer the tan-striped males. White-striped females, more pushy than their tan-striped sisters, grab the tan-striped bachelors right away, so these pairs form more quickly than the opposite combination. Males of both morphs tend to prefer the white-striped females, but those females quickly hook up with tan-striped males if they can, so eventually the leftover birds will form pairs consisting of white-striped males and tan-striped females.

https://www.audubon.org/news/the-fascinating-and-complicated...

Granted, the question of homo-zygotic white offspring viability is not even touched.

Edit: yet another article tentatively says the double-white sparrows to exist, but in far lesser number than expected, they suspect some genetic disadvantage.

https://www.cell.com/current-biology/fulltext/S0960-9822(15)...