I agree it seems odd to believe that anyone could be against having children. But the world is full of large groups of people who hold odd and unsubstantiated opinions. Perhaps he's met people who really do look down on him for having kids. Perhaps he's stumbled across a group of old fashioned Malthusians who think children = population growth = destroying the environment and he's extrapolated that belief out incorrectly onto a much wider group.
To your wider point, conservatives are absolutely victimised in some parts of society. That's what the entire article in the WSJ is about. That's what happened to Damore. Note that Damore filed a complaint with the NLRB: they wrote a memo that put the word "scientific" in scare quotes, and stated that bringing up scientific studies of gender differences was sexual harassment. So literally attempting to argue conservative viewpoints by reference to scientific studies is now considered sexual harassment and the government will not defend it: if that isn't going to create a feeling of victimisation, what is?
However, such a position is very much unusual and marginal in the US or (IMHO) the Anglosphere in general. By contrast, I had encountered some rather aggressive childfree activists on the Russian-speaking segment of the Internet.
Having children is literally the most socially acceptable thing you can do amongst almost all political ideologies. Its incredibly common and parents are held up on a pedestal just for existing.
But he thinks the liberal elites are somehow out to get him for being a parent and he's a persecuted minority. He's said "liking being a parent is such an unpopular opinion."
He has a million other benign things he thinks he's being persecuted for but the kids thing is just the most bizarre and totally detached from reality.
It's like, you'd have to totally ignore reality to have this viewpoint... At the same time it seems like a really sad life.
I wonder if they actually believe this shit or it's just a means to an end.