What would/should the crew have done when the system dropped to alternate mode ? How might they have determined what the aircraft was doing and how to respond? In fact the correct response was to do nothing, which they failed to do. It would not be hard to program a system to do better than that :-)
Alternatives? $0.1 accelerometer would have been enough ! So would GPS or simple interpolation. The system 'knows' the attitude and what the engines are doing, it knows the ground speed now and what the airspeed was a few seconds ago. Of course those do not give the actual instantaneous airspeed, but it would not require an especially smart system to detect that the airspeed data was simply missing and to use other sources of data to make an approximation that would have been good enough to keep the aircraft flying until the airspeed data stream returned. I don't know if pitot tubes have ice detection as well as anti-icing - in fact, I think if I was to design a pitot ice detection system it might look for anomalous readings or big pressure drops across critical parts!!
Of course you should now be asking how the system decides which set of data to trust if all the redundant but not truly independent members of a system drop out - eg all the pitots or all the GPS (plus both alternative GPS-like systems) which might conceivably give spurious but matching values under some circumstances. I don't think it is hard to manage such circumstances.
And of course, under some highly unusual circumstances, the system gets it wrong and everyone dies. How is that different from today's arrangement?