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Not an answer, but the closed subgroups of $S_\infty$ are classified in this paper by Bergman and Shelah: George M. Bergman and Saharon Shelah, MR 2280223 Closed subgroups of the infinite symmetric group, Algebra Universalis 55 (2006), no. 2-3, 137--173. To answer Dima Pasechnik's question in the comments, the Baer-Schreier-Ulam theorem follows from the ...
In the form asked in the edit (for every $f : G \to \Bbb C$ continuous, does there exist $x \in G$ such that $\int \limits _G f(g) \ \textrm d g = f(x)$?), the question has a negative answer. If $\chi \ne 1$ is a character on $G$ and if the statement in the question were true, there would exist $x \in G$ such that \$\int \limits _G \chi (g) \ \textrm d g = ...