Low-dimensional irreducible 2-modular representations of the symmetric group

An earlier reference for the minimality of these degrees over any field is L.E.Dickson, Representations of the general symmetric group as linear groups in finite and infinite fields, Trans. Amer. Math. Soc. 9 (1908), 121-148. This can be found in Dickson's Collected Works.

I believe that the minimality of these degrees for representations over the field of order $2$ was proved originally in the paper

A. Wagner. The faithful linear representations of least degree of $S_n$ and $A_n$ over a field of characteristic $2$. Math. Zeit, 151:127–137, 1976; DOI: 10.1007/BF01241824.

For $S_n$, the minimal nontrivial irreducible representation has dimension $n-1$ or $n-2$ for all $n$, as you conjectured. For $A_n$, there are a couple of exceptions: $A_7$ and $A_8$ have representations of degree $4$.

If you fix a prime $p$ then for $n$ sufficiently large these $n-1$ or $n-2$ dimensional representations are indeed of minimal dimension. For this I believe the right reference is "On the minimal dimensions of irreducible representations of symmetric groups" by Gordon James, although I don't have access to it at the moment.

I certainly would not be surprised if there are counterexamples for small $n$ and $p$ though.