That's a very interesting proposal, but I wonder how it would work out in practice. As all eggs have one X chromosome, that effectively creates a system three sexes: Female, Male (X-gamete predisposition) and Male (Y-gamete predisposition), which we can call F, Mx, and My. The "x" and "y" predispositions might also be heritable with some frequency ("h"). Perhaps the predisposition is not absolute, but just a probability "p" favouring one gamete over the other.
I wonder what kind of equilibrium this system would come to. For now I'll assume that females have no preference for Mx over My or vise-versa.
For reductio-ad-absurdum, let's imagine what would happen with p=1 and h=1. Then Mx fathers have only daughters, and after one generation, the population is entirely My and F. After two generations, it's only My, and there is no third generation.
Next let's see what would happen with p=1 and h=0. Again, Mx fathers have only daughters; My fathers now have 50% My sons and 50% Mx sons. Starting from a population that is 50% F, 25% Mx, and 25% My, the second generation is again 50% F, 25% Mx, and 25% My, and with all parents having an equal number of offspring. So at a glance that looks like a stable population, but we'd need to look deeper to see whether any initial small unbalance between F, My, Mx returns to that equilibrium or diverges, and also whether there's any advantage to a higher "h" or a lower "p" that could gradually cause those traits to spread. For example if some kids with a "p=0.5" gene have an advantage over p=1, then over time the Mx and My distinction will get wiped out and we'd just end up with M and F.
I'm too lazy to analyze it but it would be an interesting system to simulate!