The Black–Scholes–Merton model does assume a log-normal distribution of price changes, and the tails of that distribution are not fat enough to account for the

The Black–Scholes–Merton model does assume a log-normal distribution of price changes, and the tails of that distribution are not fat enough to account for the frequency of large price changes observed in real-world markets. But your criticism is essentially off base. The main reason the Black–Scholes–Merton model remains the workhorse of derivatives pricing nearly a half century on, and the reason two of its creators were given a Nobel prize in 1997, is that the model has proved to be enormously adaptable. One can relax the assumption of constant volatility in various ways, assuming instead that volatilty depends on the price level, on other factors, or that it is itself random. These embellishments retain the Brownian motion and its normal distribution as the primary source of randomness, but the added volatilty structure allows for almost arbitrarily fat tails and all sorts of complicated dependencies in the resulting distribution of price changes. The fields of mathematical finance and financial engineering are in large part concerned with coming up with tweaks to the Black–Scholes–Merton model that provide the complications necessary to price idiosyncratic securities while maintaining as much of the simplicity of the basic model as possible so that pricing (usually numerically) remains tractable. As JumpCrissCross pointed out, there have also been numerous follow-on models developed that make different assumptions about the basic distribution of price changes, going back to Merton’s jump-diffusion model of 1980. Most of these models can still be viewed as embellishments of the Black–Scholes–Merton model.