No, Celticfox, it wouldn't be better. There's no reason to think the feds would do as well as one of the good districts instead of one of the poor ones (that's certainly not the case in general when they so something, including education), and it's better to have a variety and experimentation and find new things that work rather than have one big mediocre system.
Yes, sometimes it's inconvenient (moving is in many ways), and maybe an interstate compact to make that process easier would even be appropriate (as happens for, for example, drivers licenses).
But furthermore, education is deeply reflective of values that change wildly as you cross the country, and each community should be able to choose how they want to do it. Nationalizing it would cause permanent war, anger, hatred, and misunderstanding. Such region-dependent and controversial things should be, where possible, handled at the state level so that disagreements can remain just that and not become tyranny of one culture over another.