I know not many people agree with me on this, but I don't think 5-6 people recognizing black badges should be such a big deal. If I remember correctly, all people we've seen with knowledge of Section 31 have been either Starfleet officers (I'm not even sure we've seen enlisted crewmembers recognize them) or high-level Starfleet diplomats and their immediate next of kin. I definitely don't think this is a critical mass that would inevitably lead to Section 31 being common knowledge. The civilian prisoner who pointed the black badges out in Context is for Kings obviously didn't know what they meant. However, it does raise the question many have pointed out before on why they even need identifiable black badges in the first place.
But it ultimately doesn't matter even if Section 31 was commonly known in the mid-23rd century. After the whole unpleasantness with Control, they'll obviously have to retreat into the shadows and with most of their equipment, infrastructure and personnel being gone, they'll have to rebuild everything from scratch. If most data on them is purged and classified, then one hundred years is a long enough time for them to descend into obscurity. How many people today do have a working knowledge of turn-of-the-century state security organizations that weren't even classified? How many people know today who the Okhrana, the Evidenzbureau or Room 39 were?