That should not, in itself, put you off Edinburgh if it is otherwise a good choice, as you will be learning about flying from the moment you approach the aircraft to check it (from before then, if your instructor does decent briefings) and throughout the transit time to and from Fife. Presumably as Edinburgh is licensed you would be logging p/ut time for the whole trip. Even if you were flying the transit legs as a passenger, and not logging the time (presumably not paying for it either, if such were the case) , in reality your instructor would probably be letting you (making you) do all the flying and, if you don't care about hours (and if you intend to be a hobby pilot hours are, in my opinion, irrelevant: it's the fun that matters, not the numbers on the page), then why not?
Having said this, if you are paying for the time spent taxiing and holding at Edinburgh, and then having to fly to Fife to do circuits, these might be factors pointing in favour of taking Fife as your base, other considerations as to school quality, ambience etc being equal. There will be days, as you approach the solo stage, and during post-solo consolidation, when all you want to do is launch straight into the circuit. You may also find that on some days the weather will permit circuit flying, but not a cross country detail, (unless your instructor fancies demonstrating to you a bit of real-life IMC during the hop from Edinburgh to Fife).
Whatever you choose, I look forward to reading your first-solo announcement and your successful skills test announcement here on pprune in due course.