I do believe Jim said that nobody knows of their existence. They must be a reasonably large scale operation. Why? Because within virtually hours of a video of a werewolf being recorded they showed up and took the evidence. That's either luck, or they are very good at locating this kind of thing. To be that good usually requires a pretty good sized network of informants an intelligence gathering capabilities.
a) The supernatural nations themselves do this type of policing - and Harry himself suggested that they were the ones that took the tape and insured people considered it a fraud.
b) Anonymity and "a large scale operation" do not go together. Any large scale operation would be known to the supernatural nations as they do monitor governments. And the library of congress' budget is public knowledge.
c) Human nations do not believe supernaturals exist - or they would act differently. If they do not believe supernatural exist, then they will not spend the billions necessary to fund a supernatural focused agency. And if they did, they would have recruited every black cat captain in the various police forces - which means Murphy would have worked for them years ago. Hell the Church knows they exist, but their organization is still a skeleton organization as described by the good father.
Why just a few hundred years? In Cannon humans have been killing wizards, and supernatural beings for a really long time. It's only in more modern times that humans stopped believing in it. Wouldn't it make sense that over a long period of time these groups grew, and modernized with the times, and what we have now is an end result of that growth?
It is a annex of the library of congress - which means a few hundred years old at most. Not saying that naturals have not been monitoring supernatural for longer, but this particular organization has probably not been around longer than that. The agency was almost certainly instigated by some people "in the know".
Keep in mind that intelligence agencies of any kind are a recent thing (few hundred years). Diplomats as a specific organization have only been around about 500 years. Prior to that (and outside the church), there really was just the nobility and the merchant princes and guilds (which is basically what the white council is). Agencies simply did not exist.