Well, where I live is just like that, but it's just outside Edinburgh. Probably a stretch too far for you?
For £650k, you'd get a massive house, with sea views, parking for several cars etc, large garden.
Ours is a small town (10k people), but 15 minutes on the train gets you in to the heart of Edinburgh, which has everything you need. But closer still is Dunfermline, there is everything on your list. Takes about 7 mins in the car, but there are excellent bus links.
But in the town itself, we have everything you need anyway. Multiple restaurants, take-aways, tennis, badminton, churches, community clubs, bowling green, Tesco, Aldi, Asda, dry cleaners, charity shops, 5 golden sandy beaches, coastal path, woodland walks.
Not in the slightest bit cliquey - I'm English, never had a problem. In fact it's so friendly, we are always bumping in to people for a chat when we walk the dogs.
Have a look at :
Aberdour
Dalgety Bay
South Queensferry (a bit dearer property wise).
North Queensferry
We are in a 5 bedroom house which sits on the coastal path, with sea views - probably worth £250k. I can see Edinburgh from my windows.
Like I say, for £650k, you'd be laughing.
I used to live in London. No way would you get me back there, or even close - you can get so much more for your money if you move North.