$\newcommand\ep\epsilon$In view of the Berry--Esseen inequality and the inequality $\sum_{i=1}^n E|X_i|^3\le s_n^2$, $$P(1+|S_n|\le\epsilon s_n)\le 1(1\le\ep s_n)\Big(P(|Z|\le\ep)+\frac1{s_n}\Big) \le 1(1\le\ep s_n)\Big(\ep+\frac1{s_n}\Big)\le2\ep, $$ where $Z$ is a standard normal random variable.
So, $P(1+|S_n|\le\epsilon s_n)\to0$ uniformly in $n$ as $\ep\downarrow0$. Thus, $P(1+|S_n|>\epsilon s_n)\to1$ uniformly in $n$ as $\ep\downarrow0$.
