The independent schools in Edinburgh can be quite large, as can the popular state schools. DS is at Watson's, which is the largest school in Scotland in terms of numbers. However, we chose it because we went on several tours and really liked the ethos of the place. Also, the junior school, senior school are on the same campus so we felt that would make the transition between stages easier. And the playing fields etc. are all on the same campus so DH felt that the children would have more use of them e.g. they wouldn't have to use some of the gym time travelling to the playing fields.
DS was at Sciennes and we moved him to Watson's. I would have to say he loves both schools, but prefers the Watson's playground.
We live in the south side, and I really like it. However, we are looking to move within walking distance of the school and it is hard to find anything we can afford. So I can sympathise with you.
I think the north / south thing for me is that the south is nearer to the hills etc. so it feels quite open. While the north is near the beaches and river so to me feels a bit more enclosed. DH grew up in the north side and feels the exact opposite to me! It's all just based on gut feeling really!
Stockbridge is great, similar feel to morningside. I don't know the other areas too well, but I hear Blackhall is nice, and Ravelston appears lovely.