Hardly "most." The territories we've seen are a tiny, tiny fraction of the respective quadrants. Look at
the galaxy map from Star Trek Star Charts. You see that little silver dot marked "LOCAL SPACE (sphere 1,500 LY in diameter)"? That's the equivalent of
the "Known Space" map from the rear foldouts of the book. That tiny dot contains the entire Federation, Klingon, Romulan, Cardassian, Tholian, Breen, and Ferengi territories with plenty of extra room besides. You can also see how tiny the space is between the Idran system and the Founder homeworld (and frankly I think the book greatly overestimates the scale of things in the Gamma Quadrant), and you can see the very narrow line representing
Voyager's journey (and keep in mind that the ship actually skipped over most of that line thanks to various transwarp jumps and wormholes and the like, so it's really just a few short, scattered dashes along that line). All the exploration we've seen in all the Trek shows has covered only a few tiny specks of the galaxy. Even if you throw in the "Approximate Limit of Explored Space" outline around the UFP and its neighbors, I'd say the total explored volume adds up to less than 5% of the entire Milky Way. And that map is based on an old assumption about the galaxy's radius -- we now think the galactic disk might actually be 60% wider, meaning it would have over 2.5 times the total area. Which would mean the explored total would be less than 2% of the whole.