You don't have to set foot in a public school to take these tests, if you're a homeschooler. You just sign up online, pay with a credit card, and pick whatever test center in your area you want. Many universities public & private serve as test centers.
That's true, However, in practice, those universities and and various other test centers may not be administering the test on the day you want/need to take it. There are many testing centers, but they do not all administer the test every time, or they can fill up before the sign-up deadline is passed. Just FYI. Sign up early and check out the various testing centers by DATE.