en.wikipedia.org
It's from the 90s, and people found evidence for it from the 50s, so it's before the worst of the woke stuff, but yes, the later on in birth order a male is compared to his brothers, the more likely they are to be gay.
It probably has to do with a combo of kin selection + lowering competition for mates. Kin selection is because a brother is the genetic equivalent of a son, and this applies even more broadly than you'd think, on the level of tribes or larger (remember, each gene combo is in competition with each other possible choice for that position, so yes, all the redhead genes are working together in some sense).
Mate competition is a good thing for the species as a whole and for the opposite sex of the competition, but a bad thing for an individual or a small group of the same sex that shares a ton of genes. In that case, the eldest child is likely the healthiest (as the parents were younger and likely more fit when they had him, and there was less competition between siblings for food during development), and thus the most likely to succeed, so siblings bowing out of the competition is a good thing.
How do they bow out? Well, they end up gay. How? Well, the mother does have higher antibody levels to a specific protein for each son they had had before, so that might be it. Note that this is an environmental cause, not a genetic cause, despite it's effect being "born gay" instead of becoming gay after birth, as the environment causing gayness is the womb.