Let $T(n,c)=\{m\in \mathbb{N} \ : \ m<n, \ \omega(m)<(\log\log n)^c\}$. Then we have

$S(n,c)\subset T(n,c)$ and $|S(n,c)|\leq |T(n,c)|$. Also, if

$T'(n,c)=\{ m\in \mathbb{N} \ : \ m<n, \ \omega(m)<(\log\log (\sqrt n))^c\}$, we have

$ |T'(n,c)|+O(\sqrt n) \leq |S(n,c)| \leq |T(n,c)|$.

Thus, we see that if $\lim_{n\rightarrow\infty} |T'(n,c)|/n$ and $\lim_{n\rightarrow \infty} |T(n,c)|/n$ exists and equal,

then the limit $\lim_{n\rightarrow\infty} |S(n,c)|/n$ must exist and equal to

that of $\lim_{n\rightarrow \infty} |T(n,c)|/n=\lim_{n\rightarrow \infty} |T(n,c)|/n$.

As commented above, we use Erdos-Kac Theorem, then we see that
$$\lim_{n\rightarrow\infty} |T'(n,c)|/n=\lim_{n\rightarrow \infty} |T(n,c)|/n,$$

and the value of the limit depends on $c$.

For $0<c<1$, the value is $0$,

For $c=1$, the value is $1/2$,

For $1<c$, the value is $1$.

Therefore, we obtain that

If $0<c<1$,
$$\lim_{n\rightarrow\infty} |S(n,c)|/n = 0,$$

If $c=1$, $$\lim_{n\rightarrow\infty} |S(n,c)|/n = \frac12,$$

If $c>1$, $$\lim_{n\rightarrow\infty} |S(n,c)|/n = 1,$$

To obtain a finer result, we use Renyi-Turan (1957) in the form
$$
\sup_{x\in\mathbb{R}}
\left|\frac1n |\{ m < n : \frac{\omega(m)-\log\log n}{\sqrt{\log\log n}}\leq x\} | - \Phi(x) \right| =O\left( (\log\log n)^{-\frac12}\right). $$

Then we have
$$
\left||T(n,c)|/n - \Phi\left( \frac{(\log\log n)^c - \log\log n}{\sqrt{\log\log n}}\right)\right|=O\left( (\log\log n)^{-\frac12}\right).$$

Then we have the following asymptotic formula
$$|T(n,c)|/n = \Phi\left( \frac{(\log\log n)^c - \log\log n}{\sqrt{\log\log n}}\right) + O\left( (\log\log n)^{-\frac12}\right).$$

To treat the $\Phi$ term, we use Chernoff bound :

If $X$ is standard normal distribution, then
$$
P(X\geq a) \leq \exp\{\frac{-a^2}2\}.$$

Now, let $\chi(c)$ be $0$, $1/2$, $1$ when $0<c<1$, $c=1$, and $c>1$ respectively.

Then we see that
$$
|T(n,c)|/n = \chi(c) + O(\exp\{ \frac{-f_c(n)^2}{2} \}) + O\left( (\log\log n)^{-\frac12}\right) $$
where
$$f_c(n) = \left|\frac{(\log\log n)^c - \log\log n}{\sqrt{\log\log n}}\right|.$$

The first error term does not appear when $c=1$. For fixed $c\neq 1$, the first error term is consumed in the second error term.

We can treat $T'(n,c)$ the similarly.