The purpose is to practice slowly with care to learn the moves thoroughly so that when it's time to perform them quickly you'll know the form and can flow quickly - the teacher was showing slow demonstrations.

I've never fought anyone, but as I've been taught the moves in Tai Chi's martial technique are generally intended to put people down quickly and hard by doing things like breaking elbows, eye gouges, and other grappling intended to severely injure the other party, essentially breaking every MMA rule that would get you disqualified.

This all varies significantly depending on the teacher, my teacher studied Chinese grappling, Mongolian wrestling, Aikido, and other arts that I'm sure he brought elements in from.

https://www.youtube.com/watch?v=vc3Q6vqvemA this is pretty much what high level skills in Tai Chi look like. You can see that it's mainly very, very secure footwork and explosive coordinated force delivered when the opponent is off balance -- this is a grappling situation but the logic is pretty similar if they were hitting each-other or even using swords / spears.