Waterproof is a tricky word that may mean different things in different contexts. In some contexts it might mean that water can't pass through from one side to the other. But it might also mean unaffected by water. Cotton can be woven "waterproof" fabrics, like canvas or sailcloth. Both handle water very well with no ill effects, but that doesn't mean they don't get wet.

I agree that the video is deceptive, implying that the saddle will shed water without getting wet, and maybe it would when new. But any amount of riding will wear off the sizing, and unless each strand of the fabric was somehow coated before weaving, once the surface is worn, it'll act like any cotton fabric. OTOH, it might breath better, and not do any of the nasty stuff that wet leather saddles might.