Some people take the importance of parenting as justification for parents to have extreme powers to make absolutely sure nothing goes wrong. This is not a freedom-loving attitude. If power over other people is bad in general, then when things are important is the *worst* time to rely on such power. The higher the stakes, the more critical it is to do things properly! The same issues come up in politics. People say courts and police are too important to leave to the market. (Why don't they say this about groceries?) That only makes sense if the government is better at doing them. If the market is better then we should be saying it's too important to keep letting government do it.
What information transfer to the next generation means to me is that this is where improvements need to happen. For an improvement to be lasting it needs to be appealing to young people. If we raise kids to know exactly what we know, and be just like us, then progress comes to a halt. That isn't what we actually do. But it may be something we do for certain issues. Children are especially supposed to adopt the religion, morality, and values of their parents, but then are given scope to, for example, believe newer scientific ideas. So scientific progress continues, but religious progress barely exists.
It seems plain to me that we should absolutely want our children to question all our ideas and not accept anything uncritically. While that is a good attitude for everyone to have in general, it is most critical for children to have that attitude towards their parents. Children have a fresh perspective. They aren't yet set in their ways. Their life doesn't revolve around any opinions they have -- changing them is still easy. Contrast with a politician or a scientist or a preacher -- if they change their mind it's a big deal. Changing your mind about your spouse is hard. Changing your mind about your hobbies, even, is hard -- you might have a decade of experience with a present hobby, and many friends who share it.
If we have mistaken ideas that we do not improve that is unfortunate. But if our children have the same ones that is much worse. The idea sticks around a lot longer. And if their children are its victim as well, and their children after them, and so on, that is just what we really don't want. So how do you avoid that? Well what absolutely will not work at all is if parents decide which ideas their children should believe according to which they think are good. Then any bad idea they have which they don't realize is bad will be passed on. And we all know we are often blind to our own flaws. Memes can evolve to make their holders blind to them. If children make their own decisions that does not guarantee a better result. It could even turn out worse. But it gives a chance of getting away from bad ideas that make their holders blind to the badness. It also respects human dignity to prefer people choose their own ideas.