I'm not sure the Federation is so petty as to define to geography based on political concerns.
Well, the division of the galaxy into four parts is obviously done on "political concerns" insofar as Earth is used as the all-important point of reference. That's politics as much as defining the zero meridian as being at Greenwich (and not, say, in Paris) is.
If this zero meridian
were rotated a bit between the TOS movies and TNG, it might just as well be categorized as "less pettily political" than the original arrangement, as the Earthlings doing the defining would be moving the center of the universe away from Earth!
Also the Cardassian Union has always been identified Alpha Quadrant, despite the fact the Federation always bad relations with them, that got worse when the Cardassians joined the Dominion.
And? Russia didn't get labeled the East because it was evil - but because it was evil
and in the east. Had Russia been to the south of Europe and North America, it would obviously have been associated with the South.
Similarly, Klingons and Romulans can be Beta villains, but Cardassia obviously cannot since it doesn't lie in the direction of Beta, but deeper into Alpha than Earth does. A different derogatory label would have to be invented for it, then.
I think a more likely explanation is Quadrants meant something much smaller in TOS
Sure, and they still mean something much smaller in TNG (in addition to meaning the four parts of the galaxy) - several early TNG episodes use the word "quadrant" in the TOS sense of "fraction of a sector". But how does that "explain" anything, or invalidate the concept of the galactic and the sub-sectoric quadrants existing at the same time in both TOS and TNG?
The explored galaxy was much smaller in TOS.
Actually, it was much larger - several TOS episodes make mention of going to the other side of the galaxy, or going to the rim (or, in TAS, the center), or being aware of what happens in all corners of the galaxy. That got severely downscaled for TNG.
Logically, we have to wiggle out of that somehow, as Starfleet can't really have been told to withdraw from the galaxy and then subjected to an amnesia machine! Perhaps we could take some of the TOS phrasings as figurative rather than concrete? But we certainly don't have any reason to think that the TOS heroes would have had less need for galaxy-spanning terminology than the TNG heroes.
Timo Saloniemi