Child support is paid to the person with whom the child resides. It doesn’t matter if that person makes more or less than the support payer. Custody is overwhelmingly awarded to women in cases where it’s disputed. So in theory you are correct, child support has zero to do with sex. But theory and practice are two different things. In theory, women like nice, respectful, sensitive men. In practice, those men end up becoming incels.
And once again this in large part goes back to the mistakes men make like leaving the house willingly prior to the divorce being finalized. Once that happens you give up a LOT of custody rights as courts view that as abandonment. Especially if they aren't regularly seeing the kids while that happens which in many cases they don't because there is no custody/visitation agreement in place at that point.
Guys want to do whatever they want and think it has no consequences, but as I said before this is a legal issue and it has very BIG consequences if you don't do things properly and protect your rights.
Basically don't make dumb decisions by not knowing the law and your rights under the law and more importantly the rights you are giving up under the law by making these mistakes.
If you MUST leave the house then the bare minimum you need to do is visit an attorney and have paperwork drawn up documenting everything and signed by both parties as to what is being agreed to temporarily in terms of visitation/custody, etc...
Other than that do NOT leave the house or you will be sorry you did in court.