gr.group theory - Symmetric groups which are not quotients of Z/2Z*Z/3Z

Yes, this is well known. In fact, there is a name for such groups - this is a (2,3)-generation property. And yes, by now there is a conceptual understanding why all sufficiently large finite simple groups have this property - the basic ideas are outlined in this helpful MathSciNet review gently explaining the major breakthrough by Liebeck and Shalev (1996). There are more recent developments in the field, both in the asymptotic direction and in the explicit construction, such as figuring out which $PSL(n,q)$ are (2,3)-generated - see papers by Tamburini, etc. - the literature is too big to be reviewed here.