The idea behind age restrictions (I think) is that they protect kids. It makes sense why we don't let kids become firefighters or join the army, and parents can stop their kids from blowing all their money right away. But then what about age restrictions that don't protect kids?
Medical Restrictions
Parents are generally allowed to restrict their kids from getting vaccines, even the ones that have been tested the most thoroughly. In which case, it's often illegal for the kid to get the vaccine on their own. This is effectively just the government (through the parent) preventing the child from getting a vaccine. I think there's a lot of cases where that's clearly harming kids instead of protecting them.
Religion
A Muslim parent whose child wants to be an Orthodox Jew can force their child to attend Muslim services but ban them from Jewish ones. An Orthodox Jewish parent can force their child to attend Jewish services but ban them from Muslim ones. If one parent is forcing their child to act more rationally, then clearly the other parent is forcing their child to act less rationally. If the government has a problem with kids picking irrational religions then it can just mandate a state religion and decide on what's the most rational; if the government has no idea what's the most rational, then let kids pick religions even if it's a ridiculous one.
Money
Under common law, a parent can take money that their child earned and then go spend it on alcohol. I think it's pretty obvious how that is not protecting the child.
Voting (maybe)
I think the logic behind this is that parents represent the interests of their kids already. But if that's the case, then the kids will likely just vote the same as their parents, and the family should have more votes since it is representing the interests of more people. In cases where adults don't represent their kids' interests, like abusive parents, then the kids are getting no representation right now and that's a problem. Letting minors above like 14 vote might be a necessary evil that fixes it. Kids can be coerced, but isn't the current system, where only adults vote, effectively adults coercing children?
Kids are also less intelligent than adults, but the goal of voting isn't to see what smart people think, it's to represent everyone's interests. We don't test adults for voting, measure their IQ, or even restrict most mentally challenged adults from voting. Maybe those tests would have unintended side effects, but based on my list so far, can we honestly say that restricting kids from having representation isn't having negative side effects?
I could be swayed either way on this, though; I think the risk might be that a small subset of adults (like teachers) may have disproportionate power if kids are allowed to vote.
Solution?
I think there's a wide variety of cases where the government should grant children somewhat more autonomy. This wouldn't be the government forcing the child to do something, just preventing their parent from making the choice for the child. There are a lot of risks from standard voting for kids, but there are also significant risks from kids having zero representation, and I'm not sure what the best way to solve that is.
