Volatility clustering, rare events - and also correlations are potentially important missing effects?
Sure. Why do people do simulations? It is often for asset allocation/risk-management -- to see if you can stomach some worse case scenarios. Sometimes it is for option/derivative valuation.
The IID draws from an empirical distribution that contains the Black Monday crash will not only pick up the occasional market crash in the simulation, but the possibility of a crash followed the next day by another crash, etc. Of course, the probabilities, while greater than zero, are quite small (although orders of magnitude greater than under GBM with typical vols).
This is important for option/derivative valuation. Without it, typically simulated option "skews" decay too rapidly relative to what is observed. However, the simple IID model or even the block-draw model is not going to generate option prices that match the market, so that's why I didn't really discuss options.
The general rule is you'd like the simulated series to behave like the empirical series.