Having read all your requirements, I think a move to just outside Glasgow would suit you really well. It's Central Belt, without being somewhere depressing between Edinburgh and Glasgow (I'm sure there are some nice places, but it seems all a bit bland/run down to me).
I'm from Edinburgh, originally, but I'm could not afford to live where I grew up (Morningside) or in any of the good catchments. Also, the public transport is a bit better through in the West (although I've not lived in Edinburgh since the trams were reintroduced, so this could be a little misleading). Still, there are so many trains that living 'rurally' outside Glasgow still means you can affordably work in Glasgow.
I live in a village to the south of Paisley (hi other Renfrewshire MNers!). There is a train station in our village, and in 20 mins I can be in the centre of Glasgow for my job. In 30 mins, I can be on the Ayrshire coast, and in 40mins or so I can be at Loch Lomond. An hour to Edinburgh as well.
There are a tonne of NHS jobs here, lots of hospitals within easy commute (I'm NHS, but not clinical).
Our house was a smidge under £300K. We have a 4 bed (modern but well done) detached house, with rural views. Primary school within 5 mins walk. High school is not the best, but realistically, probably way better than what you are facing. We can do a placing request when the time comes, or consider other options like moving or private.
We also have an au pair. Apparently, au pairs coming to Scotland mostly look for placements in Edinburgh, but it was no problem finding one that wanted to be near Glasgow. I'm not sure about further north, but I think most big cities would be attractive (and I'm using big in the Scottish sense, so including Perth, Aberdeen, Inverness etc).
Upsides: rural, community feel. 10 mins to our nearest 24 hour store. 15 mins to Glasgow Airport and then the world. Cinemas, culture and any activity you care to name easily in reach. When our kids are older, they can be independent without us running them everywhere. People are friendly, and I have several English neighbours (and being from posh Edinburgh, I almost fall into that category) who are settling in fine.
Downsides: the weather. It is definitely wetter here than pretty much anywhere in the UK. But when the sun shines, we really appreciate it. And it is only a couple of hours to the east coast oasis of sun (if you would believe my mother from Peebles). Winters can be dark and dreich, but summers are so light. And it's so close to the north of Scotland, that you can go on wee holidays up north all year and enjoy the scenery etc. Sectarianism. I've not really experienced it, but I know it is there. Still, any place has it's darker elements, and I certainly don't feel I encounter them on a day to day basis in my little village.
PM me if you want any further details. Sounds like an exciting time in your life 