>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.
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.