Simply that B-S does not reflect skew, and, in my view, causes the vix to under estimate probability of touch for puts, and at different times, overestimate and underestimate probability of touch for calls.
Not sure VIX has anything to do with BS itself. I think it is purely related to how it is calculated.
If anyone thinks that the negative correlation between option IV and the price of stocks is just an artifact of the way things are calculated, they need to get out and trade some options and get a feel for how things actually work. Options behave completely differently in a down market than in an up market. In an up market people are happy, complacent, and making money. There are plenty of OTM calls to buy, easy and cheap, which brings all option IVs - and the VIX - down. In a down market, panic quickly sets in. Nobody wants to sell OTM puts. That drives the IV of all options up, along with the VIX.
Yes, it's true that B-S doesn't reflect the skew, but the VIX calculation doesn't employ B-S, so this is irrelevant.
VIX is not actually implied volatility, it's a variance index. You do understand that a VIX futures position and an option position are two very different animals, don't you? Nobody is denying negative correlation between implied vols and the index, however looking at the level of VIX to understand true demand for convexity is mileading. I will say it again, go and look at implied vol on a single strike, say 900 and you will see that fixed strike vol is far less volatile then VIX. Most of demand/supply dynamics that you describe is already reflected in the skew and bigger portion of changes in VIX is due to changes in at the money strike. If you're trading index options (or especially variance swaps or VIX futures and options) you would know this.
It's true that fixed strike vol is less volatile than VIX. But fixed-strike vol is a poor comparison. Let's take that 900 put as an example. As the market drops, the demand for convexity increases, driving IV on the 900 put up. But at the same time, that put is now less OTM than it was, which "dampens" its rise in IV. To test what you're saying, I'd have to compare apples to apples, which would be better done by comparing the 900 put to the 880 put after the SPX drops twenty points. Still not perfect, but closer. I've never done that, so I can't say the rise in IV still wouldn't match the rise in the VIX. Something to look at one of these days.
Statistics sucks. The first moment is inappropriate against the second. It's not a supply/demand issue, but a mathematical one.
Well, what you are saying is that in yout view, current skew does not reflect the actual dynamics of realized vols (skew is cheap). If you truly believe that you should put on either a risk reversal or, better, a long VIX futures against a delta-hedged fixed strike option (assuming your pb margins that kind of spread efficiently). I generally shy away from trading the skew outright since it almost always boils down to a simple gamma or vega position when the spot moves. Trading variance (or better, forward variance like VIX futures) against a fixed strike remedies that problem a bit.