Unless I am entirely misreading the spec, VIX is the square root of the fair strike of a var swap, not the fair strike itself. That is var swap fair strike vs vol swap fair strike. VIX normally trades above fair vol strike due to the expectational form of Jensen's inequality -- very much dependent on level. I took your post to be referring to the difference between VIX and vol swap strike. I admit I found the term "VIX fair strike" ambiguous as the VIX as a fair strike of anything only holds via a circular argument. So I may have misinterpreted your original post.
Selling egregiously priced OTM S&P puts is equivalent to getting long delta. Selling 5 20 delta (front month preferable over LEAP puts) puts will be equivalent or produce better results than buying a future or ETF. On a rally you'll make money on the delta, vol implosion (vega), as well as time-decay (theta).
Well, VIX does not really trade in itself but when a proxy for it trades (for example, the strip on the futures expiration day), the level would always be above the vol swap if you were to quote it due to the effect I described. Since the VIX in question is spot, there is no question about expectation of the fair strike (which is set), but rather expectation of the terminal distribution of realized volatility. You are confusing the two different expectations and convexities here. There is VIX/VIX futures expectation vs forward expectation of fair variance (that's what you are talking about, I think). Then there is the var swap vs vol swap premium that's due to the convexity of realized variance vs volatility.
VIX white paper says in their 1st sentence that, VIX measures 30 day expected vol. of the SPX index. My main issue was, he was looking at VIX value and planning to sell very long term option of different maturity. https://www.cboe.com/micro/vix/vixwhite.pdf
It does indeed, but in this case you should not believe your eyes since the formula they use is the formula for the fair strike of the variance swap.