It's not about competition at the sperm level, gene quality as a whole. An individual is attempting to maximize its own genes. So a woman mating with a more masculine man will give her offspring the advantage of having "alpha" genes. But alpha men are not usually strong providers as they have their pick of women and so they spread their genes widely and divide their resources (or withhold them altogether). Thus maintaining the solid provider as the adoptive parent makes up for this. Of course, a man (evolutionarily speaking) has no interest in raising another man's kids. So the woman has to be sneaky with the affair. If the deceit remains hidden, this setup is by far the best the woman can do.
As far as having kids who are themselves not good providers, an alpha man's strategy is usually one of spreading his genes far and wide. If you're desirable enough, that may be a winning strategy for you.
>Alpha Males are by definition strong providers, since they get preferential access to food
That doesn't follow. The male gets preferential access, but that doesn't mean he shares that access with his offspring. And even if he did, he would still have a larger than average number of offspring thus the amount each one gets may still be lower than the beta male's offerings to his own.
Of course in practice whether the alpha is the best provider is determined by how the community is organized. In a alpha-male-with-harem setup, the alpha is the best provider. In other arrangements this isn't always the case.
As far as having kids who are themselves not good providers, an alpha man's strategy is usually one of spreading his genes far and wide. If you're desirable enough, that may be a winning strategy for you.