If $G$ is a finite group whose order is divisible by a prime $p$ and $p^r$ is the maximal power of $p$ that divides it, the Sylow theorems tell us that the number $n_p$ of Sylow $p$-subgroups of $G$ is congruent to $1$ modulo $p$ and a divisor of $\lvert G\rvert/p^r$, and that if $n_p=1$ then $G$ is not simple. With this information alone we can decide that some numbers are not the order of a finite simple group.

> How big is the set of numbers that these two facts exclude as candidates to be the order of a simple group?

By *big* here one can mean, say, the density in $\mathbb{N}$. One can also weigh numbers by the number of actual groups of each order, so as to turn the question into

> What proportion of finite groups are known to be non-simple by using only these two Sylow theorems on the number of they Sylow subgroups?

Since most groups are apparently $p$-groups, the answer to this is probably zero.