Bearing in mind it's a capital city, home to seemingly countless universities (at least one globally revered,) and the whole conglomerate that constitutes the International Edinburgh Festival, it would be odd if one couldn't assume that Edinburgh citizens are aware of their place in the world and welcome all comers.
But obviously you need reassurance on a more granular level. As a non-white person living in the city for a couple of years a decade ago, I encountered one, single instance of overt racism that was swiftly put down by the man's colleagues. And one utterly shocking episode where a student doctor attempted to diagnose my gynaecological problem as malaria. 
There was perhaps an element of surprise amongst public sector workers - the people one deals with in the normal course of organising your life - that I had high expectations of the service they were tasked with providing. If this is a general thing it may impact your interactions with state schools, health service, etc. But equally, things may have improved over the past few years.
In all areas of the private sector (I work in the Arts) people were unfailingly decent and I never felt at a disadvantage.
So I'd say that, if you can find a school with the right ethos (this almost certainly cannot mean a school with a high proportion of non-white children, as the statistics show) and are accustomed to asserting yourself ...
Tbh, if I were moving there with small children I'd be inclined to live right in the middle of the city - in the midst of a huge international cohort of students, academics and arts practitioners - and choose an independent school serving an international clientele.