We didn't even have universal public education until after the civil war. Even then it was a matter for the states and localities, none of the federal governments business. As the Constitution has not been changed to make it their business, it still isn't.
So it is with a whole bunch of things, which are not really "commerce" but which have been brought under the Federal Government's purview, most of it since the first world war, less than 100 years ago.
It's not so much that we don't need a state, we do, but we need a federal government that is strictly limited in it's reach, state governments that are somewhat limited, and much more local control of the laws which directly affect the people.
Depends on who you mean by "We" (and, to a lesser extent, what one means by "universal")
In New England, we have had universal education since the 17th Century - even girls got some.
America before the Civil War is not something that anyone should want to emulate. An economy built on slavery an conniving expropriations was a nasty thing. Sure, it was relatively better than most of the alternatives of the time, but the point isn't comparative history, is it? When folks like you suggest that the federal government has no place in education, the response is that the federal government HAS to have exerted a powerful and coercive role in education: it was the only way to force states to educate black people with anything like parity. The states didn't want to let go of slavery, and they didn't want to let go of segregation. We should remember that the REASON the federal government had to get so powerful in the first place was that people were so damned determine to hold onto enforced racial inequality that the only way to get around it was to shatter the power of the states with federal power. Had slavery simply been let go, and segregation not been put up in its place, then we wouldn't have the powerful federal government we do.
But that isn't what happened, is it?
The Constitution, as originally framed, did not say that states could eliminate the constitutional rights of blacks. It was UNDERSTOOD that the states could, and they did. And that stank to high heaven. Going back to original intent gets to some very ugly places, because the circumstances of the country at the time of the Founders was ugly and vicious in important respects, and the federal system preserved that for far too long.
This is always the fly in the ointment when folks pine excessively for states rights and the original Constitution. What states' rights MEANT, above all, was the "right" to reduce black people to slavery if the majority damned well felt like it. And that was the fatal flaw in the flawed constitutional structure of the country.
Only federal power could overcome that, and federal power did. The reason we have truly UNIVERSAL public education, which is to say that the reason BLACK people are given public education at all, is because of federal power. That's why the federal government HAS to be involved in education.