This suggestion comes up repeatedly, but I think the practical problem is that the vertical air velocity is unknown, and in turbulence, the naturally-occurring short-term variation in it is comparable to the range of the vertical component of the airspeed over the operating range of angle of attack. Maybe with an accurate model of the airplane's response to both gusts and control movement, and data about control movements together with acceleration and rotation of the vehicle, one could estimate this with sufficient accuracy, but the first thing to consider would be more (and heated) sensors (and more robust sensor-driven software, apparently, that can survive the failure of one of its sensors.)

I would guess that with GPS, an additional problem might be the timeliness of accurate-enough data. I would be interested if someone could comment on the state of the art in this regard.

In addition to pitot tubes and AofA vanes, there are various stall warning sensors that detect the movement of the wing's stagnation point as a stall is approached, and these are widely used, at least in the general aviation category. To be fair, they are also vulnerable to the effects of icing.

Update: On reflection, just holding a reasonable attitude would probably be a reasonably effective response to pitot failure, though if your sensors are failing on account of icing, you probably have other problems.