Here's a real-life example of why ground speed doesn't help much with flying the plane.
I was in a Cessna 172 on a long cross country flight. Cruising at 100 kts indicated on the gauge (at 11,500 feet, so 126 knots true). Windy spring equinox weather.
I had a radio tuned to a navigation beacon that had DME (distance measuring equipment) ahead of me. Told me how far away the beacon was, and how fast I was approaching (internal electronic calculation based on closure rate). So a fair substitute for "GPS ground speed."
I got a consistent reading of 179 knots! I.E. I had a 53-knot tailwind.
Now - if I had really been going 179 knots, that is 21 knots over the never-exceed speed for a 172. My wings should have ripped off!
If I had lost my pitot IAS indication, and tried to use that ground-speed reading to control the plane, I would have immediately cut power and gently pulled up to get my speed below 158 knots. And slower yet to get back to a "reasonable" cruise speed of 100 knots.
Problem is - my true airspeed would have dropped to 100 knots minus the 53-knot tailwind, or 47 knots. That is the 172's no-flaps STALL speed - and since the winds were turbulent, I'd have likely caught a gust and stalled and spun.
Of course, I'd have gotten a stall warning at a slightly higher speed, leaving me in the "What the heck??!!" situation of conflicting inputs - stall warning, while my one "working" speed indication still said I was doing a safe 100+ knots. Sounds a little like AF447, doesn't it?