The federation is supposed to something like 8000 light years across, right? If that's true, then at warp 7 (new scale), it would take 8000ly / 656 times light speed = 12.195 years to travel one end of the federation to the other. To put it another way, you could walk around Earth at the equator with stops and scuba gear for the water part in less time. What would be the point in maintaining a unitary government that is so vast that it takes 1/8th of a human lifetime to cross?
Warp speed must be faster.
If I'm not mistaken, there's a section of the Federation "southeast" of the Klingon Empire, where there's just a small part of it and connected to the rest of the UFP by a skinny corridor, almost as if a ship had been catapulted there at some point, explored it on the way back, and claimed it for the Federation. There might be a few more sections like that as well, accounting for the "spread across 8,000 LY" comment by Picard to Lily in "First Contact", but the bulk of the Federation is fairly well-centered around Earth. I'd have to look at Star Charts
to be sure, but that's how I recall it at present.