Actually, the statement is false in characteristic $p$. In "An Example of Unirational Surfaces in Characteristic $p$", Shioda shows that the Fermat surface of degree $n$ is unirational if there is a power $q$ of $p$ such that $q \equiv -1 \pmod n$. This implies that there are Fermat surfaces of general type which admit generically finite dominant rational maps from $\mathbb{P}^2$.

One nice thing you can do is use Borel-Weil-Bott to realize all the vector bundles coming from these Schur functors geometrically. Specifically, you can take the bundle of complete flag varieties associated to $E$. This will have a bunch of tautological line bundles on it. If you push forward these line bundles to $S$, you will get all the vector bundles coming from Schur functors applied to $E$.

The torus fixed locus is going to correspond to the locus where the map $E \to E \otimes K_X$ is the zero map. I would expect that this doesn't have to be proper in general, but I haven't thought hard about it.