I profoundly doubt that Section 31 is anywhere near as powerful or widespread as it likes to think it is. Contrary to the absurd fantasies of fiction, the bigger a conspiracy is, the more inevitable its exposure becomes, because there are more opportunities for things to go wrong, more lines of communication to be intercepted, more people who could have second thoughts or be co-opted, etc. The idea of a vast, all-powerful, pervasive conspiracy that nobody knows about is a contradiction in terms. The only way Section 31 could've lasted as long as it has and stayed as well-hidden as it has is by being small and limiting its activities.
And it's worth noting that, by the early 24th century, Section 31 has become entirely public, its archives free for Sisko and Nog to peruse. Something gave, fatally.