No, courts will always make sure both parents have custody rights because it’s about the child’s best interest, not the parents.

No, they don't. Or rather, they're not required to (individual judges can if that's their preference). Two states require the courts start from a presumption that equal custody is in the best interests of the child unless there's a good reason for it to be otherwise (and includes an explicitly non-exhaustive list of examples of such reasons), about half a dozen require that the courts "consider" equal custody, and the rest leave it totally up to the judge's preferences and biases. Kentucky was the first state to pass a law requiring a rebuttable presumption of equal custody, and they did that in 2018 (and they were fought against by ostensibly feminist women's lobby groups).

Until the 2000s, most custody was influenced by the old fashioned "tender years" doctrine and the fallout from that - basically the idea that a child needs it's mother so keeping mother and child together as much as possible was in the best interests of the child. At this point you're likely to claim this idea was patriarchy, but it became a thing in the first place because of early agitators for women who could be seen as sort of proto-feminist who were fighting against the previous standard of putting children with whichever parent could better materially support them (usually the father). It was only later that we took to the idea that material support could simply be extracted from one parent and given to the other.

The court is biased against women, not men, because that’s how a patriarchy works.

You should probably look at how the court system actually treats people based on sex, rather than just looking at your wildly inaccurate model and assuming that the map matches the territory because it's the map you like. I can go on about how and why it's an inaccurate model, and give some examples of those inaccuracies in action if you'd like, but that's a bit offtopic.

It's especially obvious in criminal courts, and especially when a man and a woman have been arrested for literally the same crime (not just the same kind of charges, but literally the same event). For example, look at the Chicago torture case from 2017 where two black men and two black women essentially kidnapped and tortured a white guy and streamed it on Facebook. The two men got $900k and $800k bail, the women got $500k and $200k bail. They eventually all took plea deals with the men getting 7 and 8 years in prison while the women got 4 years probation and 3 years in prison. This treatment wasn't some kind of weird one-off, but its convenient and illustrative because you had four people who all did the same crime together with an even sex split and a very obvious and dramatic difference in bail and punishment.