Consider the sequence $x^t\in S$ continuously depending on $t\in[0,1]$, such that $x_0^t=-t$ and $x_n^t=1/n $ for all $n\ge1$. If $f:S\to S$ has the stated properties, $f_i(x^t)$ is different from $-t$ and $ 1/n$ for all $i\in \mathbb{N}$, $t\in[0,1]$ and $n\in\mathbb{N}_+$. Therefore, for any given $i$, $f_i(x^t)$ is either in or outside the interval $(-t, 1/n)$ for all $t,n$, hence also in or outside the interval $(-t, 0]$. But here is the contradiction: for $t>0$ there should be infinitely many $i$ such that $f_i\in (-t,0]$, and none for $t=0$.

I think there is not such an $f:S\to S$. Consider the sequence $x^t\in S$ continuously depending on $t\in[0,1]$, such that $x_0^t=-t$ and $x_n^t=1/n $ for all $n\ge1$. Since $f(x^1)$ is dense, for some index, say $17$, we have $-1 <f_{17}(x^1)<0$. Therefore, for $t=1$, we have $$-t=x^t_0<f_{17}(x^t)<x^t_n=1/n$$ for all $n\ge1$. By continuity this must hold for all $0\le t\le1$, but it is impossible for $t=0$.