Plus, what benefit would there be from dividing up the galaxy along "natural" lines? On such a scale, there'd be nothing practical about those lines. Why should the Federation be limited by the contours of a galactic arm, when it's either too small to fill up the volume or too large to even notice the subtle transition from within an arm to outside it?
Being inside or outside an arm isn't really something you could notice simply by flying from star to star. And in Star Trek, all stars, regardless of whether statistically typical to the insides of arms or the outsides, are likely to hold targets of interest. If there aren't "natural" planetary systems there, ancient civilizations have apparently made sure that there will be "unnatural" ones...
Creating a zero meridian for a spherical or cylindrar galactic coordinate system is an obvious step. Using that to divide the galaxy in half would also be pretty natural, even if it had no practical applications; dividing in four is simply the more elegant, that is, more symmetric way to go. Just like East and West here on Earth have no practical significance but have accrued a symbolic political one, Alpha Quadrant and Beta Quadrant apparently have at least some symbolic worth. And, of course, the division between Alpha and Gamma is of utmost significance in the Dominion War, in almost exactly the same way as the division between East and West in the Cold War here on Earth.
^Those ships have really slown down by the time of VOY. As Vulcan was only 4 days away from Earth (TMP) which would mean a travel time of 4ly per day.
But ships in ST travel at Plot Speed.
I think that's throwing in the towel a bit too soon.
It's perfectly natural to have ships travel faster over shorter distances. That's what they do in the real world, too - dash speed cannot be sustained for any length of time (unless your vessel is fission-powered, in which case dash and cruise speed tend to be one and the same). For journeys taking years or decades, the issue becomes even more significant: a short hop can be conducted without concern to wear and tear, but as the length increases, so do maintenance concerns. Quite possibly a ship capable of doing a hundred lightyears in a day and two hundred lightyears in a week will limit herself to doing a thousand lightyears in a year, lest a pit stop be needed where there are no pits to stop at.