Andrew Snowden and I managed to finally answer this question in our paper "The Steinberg representation is irreducible", available here. As you might guess from the title, we prove that the Steinberg representation over an infinite field is always irreducible. In fact, we prove something much more general that applies to arbitrary reductive groups over infinite fields, and also allows arbitrary coefficients for the Steinberg module.
It's worth also mentioning another recent paper by Galatius--Kupers--Randal-Williams called "$E_{\infty}$-cells and general linear groups of infinite fields", available here. Their main theorem One of their results says that the Steinberg representation for $\text{GL}_n$ (as discussed in this question) is indecomposable, i.e. is not the nontrivial direct sum of two subrepresentations. For infinite fields, the Steinberg representation is infinite-dimensional, so this is weaker than being irreducible. However, I think their proof is quite beautiful and worth reading even if it gives a weaker result.
EDIT: My attention has been drawn to two earlier papers:
N. Xi, Some infinite dimensional representations of reductive groups with Frobenius maps, Sci. China Math. 57 (2014), no.~6, 1109--1120.
R. Yang, Irreducibility of infinite dimensional Steinberg modules of reductive groups with Frobenius maps, J. Algebra 533 (2019), 17--24.
These focus on connected reductive group $\mathbf{G}$ over the algebraic closure $k=\overline{\mathbb{F}}_q$ of a finite field $\mathbb{F}_q$. For instance, we could have $\mathbf{G}(k) = \text{GL}(n,k)$ as in the question. Their main theorem says that the Steinberg representation of $\mathbf{G}$ is irreducible with coefficients in any field. Xi's paper handles the case when the coefficients have characteristic $0$ or $\text{char}(k)$, and Yang's paper handles other characteristics.
