I see Sec 31 as dipping in and out of favor and official consciousness. There is no one way it is perceived.I liked Section 31's pre-Federation incarnation in Season 4 of ENT. Since then...nope.
They had a secret fleet with a well known AI during Discovery then they were taken underground again after that debacle.
Such things happen do a degree with contemporary intelligence services. For example, the CIA operates its own planes, several dozen of them and so on. They sometimes make the news with failed efforts like the Bay of Pigs or Central American fiascos in the 80s, but then tend to remain low and quiet for a time.