Nice observation! However, that doesn't fix the immediate problem. "There is a little of that wind" is very strange, although I suppose one can concoct a "repair reading" involving ill winds.
I think the other issue is that the NP after "a little of" must be headed by a mass noun (or at least by something being conceptualized as such). "Give me a little of the cake" works because you're thinking of cake as something you measure, not something you count. "Give me a little of the cat" only works if you are dissecting the poor thing.
So "a little of the wind" or "a little of that wind" only works if you can somehow conceptualize "wind" as a mass noun, which is, as I said, rather odd.