With that budget you don't need to look at tenement flats but for the record we are in a 2nd floor TF bang in the heart of Morningside on cul de sac so it is pretty quiet. Cost less than half your budget last year. We share our garden with 7 other flats and hardly ever see anyone else in it. Huge advantage - ground floor guy does the lawn, I can tinker about with the borders without ever having to mow. To let the wee one out you'd want a ground floor though and they tend to have smaller rooms to accommodate the passage, unless you can find a main door flat.
I did nearly choke on my falafel at the concept of getting loads for your money in Braids and Greenbank mind you, so we are in slightly different worlds. If I were you just keep looking at what comes up on Zoopla, ESPC and Rightmove (sometimes they are on one and not the other). You can move out of catchment and keep your place, but do think about high school catchment.
There are no shops in the Braids itself, the nearest Morningside. Greenbank also no shops but there are one or two out the far side of Greenbank in the beginning of Oxgangs. Milk, bread and sweetie type shop.
I don't think you would necessarily spend loads of time in a car if you lived in Greenbank. Buses through are fairly frequent although I agree getting through M'side is slow whatever vehicle. Depends where you want to go!! I sold my car since moving here, occasionally use DH's car, very occasionally use Car Club. We mostly walk or bus. DS is 7 and has no problem walking into the centre of town from Morningside e.g. through the Meadows, might get the bus back after a busy day.
You specifically said kid friendly and down to Earth, and for that I'd err on the Greenbank side over the Braids. DB is in Greenbank with 2 DDs at S M'side, and the nearest family to them likewise. The only people we know in the Braids are slightly posh, kids at Steiner's and seem to feel sorry for us, and awkwardly sympathise about "how hard it must be". Think that's because I work.
Whatever happens life will be lovely though!! I'm sure we'll see you in the library/at the school fair/scooting through the park or something and we'll never know.