One thing that seems not widely understood is that the assumption of normal or log-normal distributions in stochastic calculus is like the assumption of lineari

One thing that seems not widely understood is that the assumption of normal or log-normal distributions in stochastic calculus is like the assumption of linearity in most engineering fields. It's known to be incorrect, but you can do a heck of a lot with piecewise-linear model. Similarly, using a normal or log-normal assumption within a range of parameters (such as the volatility smile) is really useful. Part of the reason a Gaussian distribution is used so much is that you need a stable distribution if you want to be able to perform algebra on your random variables. The variance of the Cauchy distribution is undefined and the variance of the Levy distribution is infinite, so Gaussian is really the go-to distribution.