Go - Weiqi - Baduk

9x9 keeps people from being overwhelmed. Its the size of a chess baord, so its familiar to those people, and easier for teaching someone the basic life and death stuff. There are so many extra things you have to worry about on a 19x19, that it can just be a huge barrier to all but the most serious beginners.

Most importantly, it's faster. 9x9 games can be played way faster than a 19x19. I can't convince my friend to sit for 45min to an hour and a half while we play a 19x19, but a 10-15min 9x9 I can definitely pressure them into lol.

When teaching fundamental classes, we would use 9x9 for those who just got introduced, easier to show life and death and simple local situations (even ladder doesn't need to run far). And most simple tsumego can fit on 9x9 easy.

But we would switch to 13x13 by the time of beginner's class (30k) where more concepts can be easier introduced and understand it is not just about fighting but positions, and then 15x15 to more "advanced" starter classes (starting kyu 初級, about tdk 21kyu to 25kyu) for practice which whole board situations would start to make sense. And finally switch to 19x19 when they are about ddk (16k to 20k).

Mind you thought this is for kids' Go class, not for adults. There is something can be said about teaching adults starting from a larger board, like 13x13. Or just starting from one corner (one quarter) of the 19x19 board, without switching to different boards. And they would get a more intuitive concept of local and whole board positions (like they are playing four 9x9 games on a 19x19, just open toward the center, and you don't have to win them all to win a game, losing a battle can still win you a war)

I think people should play 9x9 and 13x13 for a while. There's less stuff to keep track of so it makes it easier to build an intuition for the game. Plus you can play more games - if a game is played many moves after someone is winning on 19x19 that probably has limited educational value, and this may happen with less experienced players as they hope for the other player to make a mistake. (Which often happens, but they may not be able to recognize it as such due to the board size.)

The skills developed on 9x9 are VERY applicable to a real game. I have been playing for 3 years and I still study life/death and endgame. I played only 9x9 until I reached 18 kyu on OGS. And then from 18kyu ask the way to 12 Kyu, I dominated in the corners but I had to learn the big board concepts during that time. Nothing wrong with doing it this way.