Well, if homos were actually “born that way” they would never be able to reproduce and the trait would die out.
Not necessarily. Recessive traits tend to remain in a population, since it takes two carriers of the trait to produce an offspring with the trait. So, if a trait is carried by 1% of the population, the chance that two of them will get together is 0.01x0.01=0.0001, or 0.01% (1 in ten thousand). Out of their children, one will be completely normal, two will be carriers, and one will have the trait. So that trait is not getting "weeded out."
Another factor in the persistence of homosexuality is the societal pressure to have children. In the past, this meant that homosexuals were pressured into marriage regardless of their orientation. These days, they don't get married, but use means of conception that do not require the physical act. The result is that the trait is preserved.
I'm speaking as if homosexuality is a recessive trait. The dynamics if it is dominant or impartially dominant would be slightly different. This is all hypothetical; the actual biological situation is probably way more complicated and nuanced.