It's worth bearing in mind that as education is primarily a state responsibility, the vast majority of education spending is at state and local levels. Libraries also tend to be budgeted for mainly below the federal level and thus mostly don't show up in the federal budget. The Politifact article you have so wisely provided speaks solely and exclusively to the federal budget.
You're absolutely correct that education spending at the federal level is ridiculously low! There just might be an important qualifier to that.
As a result, it might be possible that statements such as "Education accounts for 3% of all government spending" might benefit from the wonderful opportunity to become more congruent with salient facts.
You're absolutely correct that education spending at the federal level is ridiculously low! There just might be an important qualifier to that.