I see a lot of child-harming policies advocated by people who openly hate children, actually.
OP is absolutely correct that a lot of people (especially politicians and the like) who profess to "love children" actually love parental authority over children, and support and enable parental abuse of children. Whereas people who "hate children" seem to more often (not always!) direct their hatred toward parents rather than children themselves.
But two things can be true:
Children have distinct interests and rights separate from those of their parents, and treating children's interests (and "loving children") as synonymous with parents' interests facilitates absolutely horrific child abuse by parents, and
Children's interests and parents' interests are closely intertwined, and policies that make it harder for parents to raise children are also harmful to the children themselves.
So, some of the things people who "hate children" do/advocate that harm children are:
Undermining the few meager protections for parents and children that exist in the U.S. (I can only talk about the U.S. because I'm less familiar with other countries) by demanding equivalency for pets or childless adults, like "If landlords can ban pets, why can't they ban children?" (because advocates worked very hard to ban housing discrimination against children; landlords would absolutely ban children if they could) or "Why should parents get time off work, what about me-ternity leave?" (the U.S. is one of the few countries in the world that doesn't have universal paid family leave, btw) (yes, yes, everyone should get paid time off, and, also, parents should get special paid time off for parenting) (yes, "me-ternity leave" is a real term, that's not a weird strawman I made up, even though it sounds like one).
Promotion and normalization of really gross, eugenicist rhetoric around poor parents, young parents, and disabled parents, or anyone who might be perceived as having children "irresponsibly" or having children "they can't take care of." The most radical Marxist who ever sang The Internationale will suddenly sound indistinguishable from Ronald Reagan as soon as the topic turns to "people having children they can't afford."
Deep misogyny against mothers, from the overt ("fat with saggy boobs, ew") to the pseudo-feminist ("It's so sad that she gave up her mind/career/identity because she was brainwashed into becoming a breeder").
A general refusal to acknowledge child raising/care as valuable labor that benefits society and is worthy of respect and, more importantly, compensation.
Advocacy for the total exclusion of children from the public sphere, or assertion of some sort of right to never interact with children in any capacity.
Relatedly, advocacy for a strict age-segregation of society, and an intense stigma on adults socializing with children. It goes very quickly from "I, personally, don't want to socialize with children" to "And why would any other adult want to, unless they're some kind of creepy pedophile?" to "Actually mixed-age socialization is bad for children, we're doing them a favor by keeping them hermetically sealed from adult life."
Either a disinterest in or opposition to including the right to parent as well as the right not to parent as aspects of "reproductive rights" or being pro-choice.
Okay, this one is pretty trivial compared to the others, but I'm tossing it in: Appropriation of children's entertainment. It's fine and good for adults to enjoy entertainment aimed at children, but it's absolutely not okay to make children a secondary audience in children's entertainment spaces. I don't agree with "I shouldn't have to deal with kids at weddings, restaurants, other adult-entertainment-focused places" but sure, I can see how you got there. "I shouldn't have to deal with kids at Disney World" where do you even get off? It's for them! It's not for you!
Those are just off the top of my head.
I see some people saying things like "When I say I 'hate children' what I mean is that I don't want to have children, not that I have any ill will toward children" but, okay, is that what "hate" means when applied to any other demographic? Do you go around saying "I hate [demographic], I don't want to ever interact with them or even hear their annoying voices, but I don't have any ill-will toward them"?