For the sake of argument, let’s assume fathers and mothers are equally important and equally competent at raising children.
There are a variety of ways a child can be raised by one parent: divorce, death, and abandonment.
We will also acknowledge the fact that, all things being equal, the mother will get primary custody because the courts favor them in a divorce. Because of this, even with our first assumption of equality of the sexes in all other ways, the average child who is raised by a single parent because of a divorce will be better off with the father, simply because of the sample bias.
With our assumption of equality between the sexes, the child should be equally well off with either parent if one dies. A child raised by a single parent because of the other parent’s death should have a better chance of being raised well than any other single parent, because selfishness and/or bad judgment has been not been shown by the parent raising the child, unlike most cases of divorce or even abandonment, where a majority of single parents chose to procreate without sufficient prudence.
It is also clearly far easier to abandon a child as a man than it is as a woman because of the 9 months it the child spends in the womb. A child abandoned by one of its parents is also the one which is probably in the worst situation of the three. Although, with our assumption of equality between the sexes, a child would not be better off with one abandoning parent than another, there will be more children in this worst situation with mothers than there will be with fathers, simply because it is easier to abandon a child as a man.
If our first assumption of sexual equality as parent is true, children of single fathers will be better off than children of single mothers, not due to some inherent superiority of fathers (which we assumed to be false anyway), but simply due to sampling bias.