Well, how big an error in airspeed would be significant? If you fly an approach at 70 kts and that's 1.3 x the stall speed, then an error of 25 knots could be catastrophic. Can the wind at circuit height exceed 25 knots? Sure... Perhaps quite a lot of the time, depending on where you fly.
On my first solo I somehow checked and set the auxillary static port open and the airspeed indicator over-read by 10 knots, so I was flying my approaches way too slowly. I really didn't like the feel of the aircraft, so I added an extra 5 knots for good luck and asked my instructor to take a look later.
So, if my airspeed indicator were to fail and I were to use the GPS groundspeed measurement to fly by the numbers I figure I would crash quite a high proportion of the time, I feel confident that I could land a PA38 without instruments - at least on a longish runway where a little extra airspeed would do no harm.
I see that you can calculate windspeed by flying a circuit - but this will only be the windspeed at circuit height. By the time you get to final approach the wind may be doing something very different. However, I can see that it might be useful for setting the trim. And for navigation with a GPS you shouldn't really need to know your airspeed - just groundspeed, position and endurance, which you can work out from rpm.
So using a GPS to guesstimate airspeeds could prove much worse than nothing, at least for landing and take-off.