I think you are mixing up a few things. 
- As @noob2 pointed out, there is a negative correlation between price changes and IVOL changes (what explains this is not 100% clear in the [literature](https://faculty.fiu.edu/~dupoyetb/return_vola.pdf) - however, a simple explanation is that markets fall much more quickly than they rise). 
- The largest positive daily percent changes in S&P500 were all in times of turmoil and declining markets. This may sound surprising but is commonly referred to as [volatility clustering](https://en.wikipedia.org/wiki/Volatility_clustering). The link has some useful details. It is again an implication of markets falling usually much more quickly than they rise. 
- VIX is computed like a variance swap (all options across the strike spectrum). For equity (indices), the surface is skewed towards OTM puts (higher IVOL for OTM puts compared to OTM calls) most of the time (if not all times). 
- I looked at the Vol surface of Gamestop during the days surrounding 27th of Jan and 10th of March 2021 and even in these extreme "greed or bullishness" times, OTM puts were MORE expensive (higher IVOL) than OTM calls. 
- Ignore at the money options (or that the price of calls is driven up). Your observation holds a lot more general. You can look at  P.409 chapter 19 [“OPTIONS, FUTURES, AND OTHER DERIVATIVES - John C. Hull: 8th edition”](https://www.pearson.com/us/higher-education/product/Hull-Options-Futures-and-Other-Derivatives-8th-Edition/9780132164948.html). In plain English, IVOL from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity. A deep OTM put will be a deep ITM call. If you have the same strike and same tenor, theoretically constructed Vol surfaces will show the same IVOL. You can have a look [here](https://quant.stackexchange.com/a/63662) for a more "formal" explanation. This need not hold with listed options, as prices (especially in illiquid markets) can be quite erratic but the general logic is the same - OTM put IVOL will be higher than OTM call vol. 
- Implied Volatility is a measure of (forward looking) uncertainty. In terms of VIX and S&P500 it seems very unrealistic to assume a scenario that is remotely related to Gamestop. 
- Even if S&P500 would be expected to have swings similar to Gamestop, OTM puts will still be relatively more expensive, which should in my opinion be interpreted as (permanently) increased demand for puts. 
- Last but not least, I had a look at Gamestops Put/call open interest (all option contracts that have not been closed, liquidated, or delivered) ratio. It went from roughly 1.05 at the beginning of 2021 to a peak of ~6.8 at the end of January. That means, no matter how much greed or bullishness there may have been, demand for puts actually dwarfed the demand for calls during this time. Again an implication that (perceived) risk or fear matters most.