This is just an ``asymptotic'' answer which add some supports.

Let $T$ be a maximal torus of $\mathrm{GL}_n(\mathbb{F}_q)$ (I am following the terminology of Carter's book), let $\theta\in \mathrm{Hom}(T,\overline{\mathbb{Q}}_{\ell})$, and let $R_{T}^{\theta}$ be the corresponding Deligne--Lusztig representation. Suppose $T$ corresponds to a partition $P$ of $n$.

It is known that, if $\theta$ is in general position (i.e. not fixed by any non-trivial elements in the Weyl group relative to $T$), then $R_{T}^{\theta}$ is up to $\pm1$ irreducible. Also, every irreducible representation appears in some $R_{T}^{\theta}$. Moreover, for fixed $n$ and $P$, when $q$ tends to $+\infty$, the ratio of the number of the general position $\theta$'s by the number of all the $\theta$'s tends to $1$. Finally, the dimensions of the $R_{T}^{\theta}$'s are integer coefficient polynomials in $q$.

The above properties imply that: There is a finite set of polynomials $D(n)\subseteq \mathbb{Z}[x]$ such that, when $q$ tends to $+\infty$, the ratio of the number of irreducible representations of $\mathrm{GL}_n(\mathbb{F}_q)$ whose dimension is the evaluation of an element of $D(n)$ at $q$ by the number of all irreducible representations tends to $1$.

All the above assertions should be in Carter's book mentioned in Professor Humphrey's answer.