I would like to show that the only proper and non-trivial vector subspaces of $\mathbb{C}^n$ that are invariant under all permutations of the coordinates are $\{(x,x,\ldots,x) \in \mathbb{C}^n:x\in \mathbb{C}\}$ and $\{(x_1,x_2,\ldots,x_n)\in \mathbb{C}^n:\sum_i x_i = 0\}$.
Can you please provide a hint to start rather than a full solution? I would like to avoid using results from representation theory if possible.
Here is a way (very vaguely, so you can do the main part by yourself):
It should be clear that you are almost done, if you can show the following: If the vector $v = (1,-1,0,\ldots,0)\notin M$, then $M=\operatorname{span}\{(1,\ldots,1)\}$. And this you can show as follows: If $v\notin M$, then the transposition $T_{12}$, restricted to $M$, is the identity on $M$ (you can prove this by looking at eigenvalues and eigenvectors). Thus, the first entries of vectors in $M$ must be equal. Now, transpose a little further and you'll see that indeed $M=\operatorname{span}\{(1,\ldots,1)\}$.