In single pilot IFR, an autopilot is often your best friend. It's exactly like you say - when you're busy with everything else you want the plane to fly itself. Isn't that problem already somewhat solved in a sense? Or are you referring to Garmin Autoland (or similar) in emergencies?
By the way - I can totally see how a great GA fly-by-wire system is an improvement to maintain positive control of an aircraft at all times. I'd personally love to give it a try and see how it reduces pilot effort while flying.
It's solved for the "cruise in a straight line" sense, but for emergencies or much more dynamic situations like a final approach, autopilots don't really help all that much. We can set up our VNAV and everything, but the second we get an unexpected clearance because something changed or we need to be diverted for a faster airplane or something, you need to be a real pro at using the autopilot to quickly adjust, or hand fly. We're offering a third solution--blend the two into one unified control interface.
By the way - I can totally see how a great GA fly-by-wire system is an improvement to maintain positive control of an aircraft at all times. I'd personally love to give it a try and see how it reduces pilot effort while flying.