For the problem below, I'm wondering if we could've proved it by considering that there must exist the same remainder for $2$ different terms instead of sums. There are $n$ terms, each are subsets, and if none are divisible by $n$ then there must exist $2$ terms with the same remainder mod$n$. Then just subtract them mod $n$ would give a difference with remainder $0$ mod $n$.
You are right about the existence of two terms with the same residue modulo $n$. The only flaw in your approach is that unlike $s_k-s_l$, the term $a_k-a_l$ is not necessarily a sum of some subset of $\{a_1,...,a_n\}$.