I know everyone is jumping up and down about *A, but I'd say give KLM a fair chance. I find Schiphol perfectly pleasant when one has status... KLM also has about 5 daily flight to AMS from LCY (which is a big plus in my book). In terms of redemption, KLM lets you redeem on DL to the US (direct flights), and adding a 45-minute hop to CDG or AMS on the way to a long-haul destination should hardly inconvenience your family too much.

By the way: unlike on BA where you get charged Avios per segment, FlyingBlue (KLM/AF) charges from origin to destination, so AMS-JFK costs the same number of miles as LCY-AMS-JFK, and the same number of miles as LCY-AMS-JFK-ORD.