Saw Handle Puffs Air with Each Stroke to Blow Away Sawdust

You have more control over the saw when pulling back, then you do pushing forward.

I get it. I just don't think it's necessary. To combat this problem myself, I would start with a cut across the top, right on the line. That way there is no need to see the drawn line (because it will be replaced with the first cut). It also promotes a straighter cut in the end.

Hey Jed, can I borrow your puff saw?

Or you could just blow on it like us non-idiots would.

Western hand saws (not native to North America) do cut on the push stroke; however, a little dust sometimes does appear on your cut line, but I can't see how this device would help in any way. BTW - western saws in no way have less control than a Japanese pull saw - or you've used a cheap or dull saw.
Still, I can't wait to see this lampooned on the woodworking forums. Chris Schwartz will surely have this on his Popular Woodworking blog in 3, 2, 1......

"Or you could just blow on it like us non-idiots would."
^^^^
This.

Although I'm sure you could make a small fortune if you made a similar mechanism to a western saw; having the most expensive tools is more important than having the most reasonable tool for a lot of people...

Looks like a well equipped shop, why didn't he use the table saw?

The biggest difference between a pull and a push saw is the thickness of the blade, and thus the width of the kerf - the slice of wood that's getting reduced to dust. Push saws have to be thicker to take the force of the push without kinking or breaking, and so have a wider kerf. The table saw even more so, which is perhaps why he didn't use it.
Those delicate pull saws are at times a bit too delicate and nearly impossible to set and sharpen, effectively making them disposable. I wouldn't give up my hardy push saw, even though I like the pulls.