I once had a 30 minute argument with Melody Lauer in the back of a taxi at SHOT whether or not teaching stance to new shooters was important.
And like many things, the context that we're dealing with is important. When I talk about stance, I'm not just talking about where to put your feet, I'm talking about the overal structure of how your body interacts with the pistol, all the way up to your forearms and head/neck position. I think it's important to teach people how to do that in a way that's optimized around how the human body actual works, and not introduce artificial muscular tension (the Weaver push/pull).
But, stance falls apart when people get into dynamic shooting environments like USPSA or actual gunfights. When I'm shooting on the move or shooting around a tight fault line, you won't see me using any kind of specific stance, because I'm interacting with the gun in way that's dictated by my environment. However, because I have that fundamental structure trained into me, even if my foot position or torso position is compromised, the rest of me is solid.
On the third hand, stance becomes extremely important again at the high levels of performance when you're taking specific pistol skills such as Steel Challenge or Bianchi Cup. Scott actually talks about "binding" - where you use your foot position to help control your index on the draw, and when I was shooting a lot of Steel Challenge, I knew a lot of high level dudes that would point their foot at the stop plate on stages like Five to Go. Personally, a slight change in my stance helped me get over one of the humps on my coin run, so I think stance can matter quite a bit.