Your idea of a backup system is an excellent one - and it's already been done, in a much more reliable way! The problem with the GPS + windspeed from some ground source is that:
a) It assumes the wind at the ground station is near the wind speed and direction of the aircraft. This is almost always a wrong assumption, as the wind speed rapidly increases with altitude, and tends to cant in quite a different direction as well. For a sensor that has to be accurate within a knot or two, this is completely inadequate.
b) It assumes the GPS gives an accurate speed second by second. This is again, a poor assumption. GPS fundamentally determines position, and speed is derived from this over time, meaning it typically indicates how fast you were, a few seconds ago. For the second by second resolution you need, this is inadequate.
c) It assumes that a way, in real time, of transmitting this information, deciding which ground stations to use, what interpolative formulae to use, mixed with the GPS data, can be reliably done.
So, it's not that it's a bad idea to have a backup, just that this particular implementation of one has many features that would render it useless.
How it's actually done is as follows. On transport category airplanes, it is required that you have three independent sensors to detect airspeed. Almost universally, this is implemented the same way; a pitot tube to detect ram air pressure, and static ports to detect static air pressure. They generally have three pitot tubes, each with its own independent plumbing, all electrically heated, and hooked up to 1 of 2 independent boxes (called air data computers) that reads the two pressures and calculates the indicated airspeed, which is then signaled on two independent airspeed indicators, one on the Captain's side, one on the F/O's. The third is hooked up to a set of standby gauges that are meant to break ties in a hurry if one of the primary sources goes wonky. The static ports are easy to spot (except on American Airlines, for reasons that will be obvious), because they carefully don't put any paint near them, so you can spot them on the sides of airframes as metallic ovals, with a black paint around the edge of the oval as a 'warning', and a few small holes in the middle where the static air pressure is equalized with the outside air.
On the 737, if you look down and to the left just when you board it on a jetway, you will notice one of these pitot tubes, along with an angle of attack vane perilously close to the jetway 'gasket'. The pitot tube looks like a little forward angled wing with a tube extending along its outside edge and forward, usually with a nice, bluish sheen to the metal from being kept well above the boiling point most of the time to keep water and ice far away. The AOA vane looks like a backward slanted wing, typically cocked at a crazy angle on the ground as there is too little airflow to make it align with the wind.
One of the preflight duties of an F/O is to assure that these tubes and sensors have not been messed with, whether by an overzealous TSA inspector using them as a hand and foot hold and breaking them in the process (yes, this happened), a jetway driver snugging up the the jetway a little off the spot and mashing them (happens often), aircraft washers forgetting to remove the covers put over them after cleaning the exterior (this
killed 70 people once)or a baggage cart smearing up against it and driving off, hoping nobody will notice they did it (yup, this one too!) Check out this article on
Wikipedia about it - it's fairly good and pretty informative about why it's such a fundamental indicator and why pilots get cranky about it being at all abstracted away from the raw data!
:-)