still seems off...
I mean at the end of TOS, they'd barely mapped a 1/4 of the alpha quadrant... given the Enterprise's top speed back then, they would have mapped a tiny fraction of the alpha quadrant, I seriously doubt 800+ individual species existed around their tiny area of exploration...
I suspect the Federation has maps for at leasta 1/4 of the Galaxy. After all in the DSN pilot, the Runabout was able to identify the star Idran based on data from the Quadros 1 probe. So that system could be considered to be mapped.
Though I take it you mean explored with manned starships, but if we take the 800 figure from the novel, set circa 2369. That would be around 200 years of Human space exploration. Once you start to factor in increasing speeds of ships, greater numbers of ships a large volume of space could have been explored. Now of course some of those 800+ could have been contacted prior to the development of not contact directive. And some contacted races might not fall within the region of space claimed by the UFP.
So it is plausbale