I still do not see the complete picture. Why is the assumption of normality important? Other than to simplify the calculation of parameters in ARMA models.
The short answer is I don't know. However, the convenience of simplifying the calculations is, in my opinion, a win. Again, I am myself not convinced. Probably, my premise on the ubiquitousness of normality assumption on the residuals in ARMA models is itself flawed.
In a way, I am trying to link it with the central limit theorems. But, I may be woefully lost. I am just a novice in this.