Traffic can be a nightmare peak times. Agreed re Fulford as best school – catchments. Included in that area is Copmanthorpe, Bishopthorpe & Heslington which are all easier commute and cyclable, but the house prices are silly.
We moved to Haxby a year ago (Rowntrees School which is fine and relatively new build), busses are cheap and plentiful, 20 mins to station by bus/car +15 peak. Its bigger than a village but feels villagy still. All the villages outside the ring road I’ve visited are nice, lots of sustrans routes to choose from.
I cycle same distance to the center for work in 25 mins practically all off-road cycle track down the chocolate factory dis-used railway. On sunny mornings I cycle along the Ouse. Next door is up for sale (although optimistically at 50k more than we paid).