Let $S_\infty=\bigcup_{n=1}^\infty S_n$ be the infinite symmetric group consisting of permutations of $\mathbb{N}$ fixing all but finitely many elements. Then $\{e_i=i\}$ satisfy your conditions but no element $i\in \mathbb{N}$ is fixed by $S_\infty$.

Let $S_\infty=\bigcup_{n=1}^\infty S_n$ be the infinite symmetric group consisting of permutations of $\mathbb{N}$ fixing all but finitely many elements. Then $\{e_i=i\}$ satisfy your conditions but no element $i\in \mathbb{N}$ is fixed by $S_\infty$.

(Sorry, I see that Andreas Blass already gave this counterexample. I've made this answer community wiki.)