Teachers aren't the idealized constructs they often use to teach.
That is a completely different issue, classrooms have a specific exception (USC title 17 chapter 1 § 110 (1)) which allows "performance or display of a work by instructors or pupils in the course of face-to-face teaching activities of a nonprofit educational institution, in a classroom or similar place devoted to instruction".
The issue at hand here is that textbook publishers can't (or won't) include the speech or even major excerpts of it. And although a teacher can buy the speech DVD and play it, it's not a very good support to study the speech.
Furthermore, use of the video is much more heavily controlled by the King estate.
We still have Norton's Antologies, right? Many of those are 100% filled with copyrighted works they need to pay royalties on. And they do. And everything works out just fine.
The issue is the price not the payment per-se.
Here is even an illustrated form for children with a CD: http://amzn.com/0375858873
Showing your age? :)