I grew up near Chester and agree it is beautiful. If you follow the advice of AGAA4, nearby villages include Mickle Trafford, Frodsham (market town), Kingsley (very quiet and has wonderful views over the Cheshire Plain), Alvanley and Manley. Sadly the smell of the oil refinery now reaches Helsby. Further afield is Daresbury, its church windows depicting the works of Lewis Carroll. Runcorn and Warrington, massively extended as new towns, have excellent road links.
I moved from Delamere Forest (very attractive but not where my future lay) many moons ago so can’t comment on golf courses.
As your daughter lives in Cheadle have you considered Lymm? It’s in Cheshire, (further from Chester), has a village feel, might satisfy some of your criteria and is 15.8 miles from her. There’s Lymm Golf Club too for your DH. Pretty canal, barges, some niche shops, a mix of houses.