re: Why is there school today but not on MLK Day?

MLK day was started as a kind of... thing to make people feel better and feel like the government was more respectful of certain issues. Similar to the adding of a MLK street in every major and minor US city.

There are a lot of holidays we get off for and others we don't.

But the real honest to goodness reason is simple logistics.

You have a heck of a lot more off days in the Fall semester than you do the winter/spring semester. It's easy to take away a day in late winter as there is no thanksgiving or Christmas holidays, just spring break.
So the main reason why is that there are too many off days in the first half of the school year.