Pretty straight forward and to the point, right? So simply point to the article in the Constitution where the Fed is supposed to fund education, and we'll be on the same page.
Otherwise, it's completely up to each state to tax and spend how they see fit. Any state that wants can make the choice, and I think it's a great system where we can freely move to any state which taxes and spends to our individual liking. If only the Fed could ease up a little bit, rather than still running a deficit while collecting more money than they ever have before even in inflation adjusted dollars [1]...
Aside from the technical considerations of the Constitution, if you take a quick look at the New York Fed published slide deck [2] on trends in college loans, you can learn a lot about how much the Fed is already subsidizing the college industry and "footing the bill" indeed since a large percentage of these loans will never be repaid. They can't ever be repaid, because too many recipients of these loans don't actually gain anywhere near the marketable skills necessary to cover the cost of tuition.
[1] - http://www.cnsnews.com/sites/default/files/imagecache/lightb...
[2] - http://www.newyorkfed.org/newsevents/mediaadvisory/2013/Lee0...
I don't know about "supposed" to, but the federal government is certainly "able to" fund education.
There is the General Welfare Clause that affords powers to tax and approriate revenue:
to pay the Debts and provide for the common Defence and general Welfare of the United States;
And of course, the Necessary and Proper clause:
The Congress shall have Power ... To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof.
Ultimately, however, the Constitution is interpreted by the Supreme Court, not lay-persons, so you really can't just say that it's "straight forward and to the point" without addressing any case law on the subject.
I'd be open to free education for all if the schools would pare back on administration and extravagance (since there's a strong argument that taxpayer dollars can be well-spent educating a workforce, but not a strong argument that taxpayers should be funding educations at the equivalent of an all-inclusive resort). I'd be even more open to taking the federal financial aid budget and distribute it to schools with a mandate that it be distributed in the form of grants and financial aid, so that in order to maintain their quality of life, they'll be forced to work to actually generated positive outcomes for all of their students, rather than following and promoting the successes and ignoring the rest.
These are serious questions.
(*insert caveat that it's hard to really compare national governments when a single US state can have a larger population to manage than an entire European country)