My hypothesis is that clip of Teahupoo leaves you breathless because it increases your sense of presence.
Your mathematically inclined mind can grapple the complex geometry and fluid dynamics enough to get a more visceral sense of what is actually happening from these two-dimensional film clips better than someone in the general population. You haven't seen a bunch of this because the barriers to true appreciation of what's going on are rather high, especially for non-surfers.
Even though you don't surf, I think you might enjoy this breakdown of what's going on in a breaking wave on pages 3-7.
Something like this happens in all waves, at all scales- Teahupoo is just the most vibrant and intense.