Enterprise explains that.
Enterprise. That says it all to me, but anyway, I won't take the easy way out and I'll continue on.
All of the regional powers either had been at war with each other, or basically existed in a state of Cold War. Humanity was the only faction trusted by all sides, and became the glue that pulled everyone together to form the Federation.
Also, the name “Earth-Romulan War” implies that Starfleet took the lead in fighting off the Romulan incursions. Therefore, Earth’s importance in both providing defense and leadership probably gave it much influence in setting policy when coming up with and implementing the Federation.
Earth, Vulcan, Andor, and Taler don't make up the entire galaxy. I'm sure the Cardassians and the Klingons didn't give a shit about all four of those powers fighting the Romulans. After the Romulan War, when the Federation is actually founded maybe, but not during, when it looks like four separate powers that could go their own way at any point. Even after the Federation is founded, it's an upstart, and those outside of it wouldn't agree to changing how the Galaxy is mapped out to accommodate them.
It works better if there was some type of galactic agreement long before Earth or any of those other powers came into the interstellar picture, but it would be something they all went by anyway as they all became spacefaring. Because if you're dealing with aliens in space, you have to use the most common lingo so everyone has a common frame-of-reference. The 22nd Century isn't "early days" for everyone else, it's only "early days" for Earth and the Federation.
Or, on the other end, everyone could've had their own definition of a quadrant, until all the established galactic powers agree to a standardized system after the TOS Movies but before TNG. By this point, the Federation isn't some scrappy upstart, and it's dealing with a lot more of the galaxy.
EDITED TO ADD: Anyway, over and over again, on DS9, Sisko and other characters talk about how they're in the Alpha Quadrant. All throughout VOY, they're talking about how to get back to the Alpha Quadrant. If Earth was in at the Alpha/Beta Quadrant Boarder, they'd be talking about the Alpha
and Beta Quadrants. In TNG, DS9, and VOY, whenever the Beta Quadrant is referred to, it's deep space that's far away
but, unlike the Gamma Quadrant and Delta Quadrant, is still within reach.
If we're talking about TNG/DS9/VOY, I don't go by what was said in TOS or the TOS Movies, when "quadrants" were referred to
very differently, and didn't mean what they'd mean later. That includes TUC. I think Leonard Nimoy and Nick Meyer misused the system that was set up in TNG. Neither of them was really that familiar with TNG beyond the fact that it existed, they extrapolated what they heard from whoever they asked about it, and the TNG/DS9/VOY writers ended up doing something else.
Then comes Manny Coto who thinks, "But I have to explain this!" No, Manny, no you don't. This is
one of the things I don't like about prequels, especially when they're done badly. And, it has to mentioned, Manny Coto was a conservative. Some of those world views of his had to have seeped into his writing. The Federation is an analogue for America so, of course, everything has to revolve around the Federation, with Earth being right at the center of it.