Sure, but those math profs have probably taught the same course 20+ times. Going through material that many times will permanently burn it into your brain.
Absolutely. And in mathematics, advanced structures and theorems are built up layers by layer upon more elementary material. A professor who has mastered presentation of undergraduate material on a topic also likely teaches a graduate course on it, and mentors students on it, and does research on it.
They can talk about their chosen topic at many levels to many different audiences, from general audience (who may provide funding to them), high school students (outreach and recruiting), university students, and peers. This flexibility is an important part of being a very successful mathematician, and you have to burn it into your brain to reach that level of fluency.
I took a philosophy class with a professor who had taught it about that many times. Talked to a guy who had taken it before me, he said that the prof has literally word-for-word memorized certain parts of the lectures because he's figured out and internalized the wording he thinks is best. Mathematicians can certainly do the same.
They can talk about their chosen topic at many levels to many different audiences, from general audience (who may provide funding to them), high school students (outreach and recruiting), university students, and peers. This flexibility is an important part of being a very successful mathematician, and you have to burn it into your brain to reach that level of fluency.