The pitot system is cheap, simple and well understood. The AF447 system iced up due to the pitot heating system not being powerful enough to cope with the prevailing conditions at the time.
So maybe it is not that simple, and especially not fully understood? Pitot has always and will be vulnerable as it is exposed to anything you may find in the air, hail, rain, birds, insects, ice crystals... Pitot may fail in an undetectable way, it is not always obvious that it is
not working (Birgenair is probably not the best example, as this was even detected, but ignored as it later on seemed to work).
A backup airspeed and Mach could simply be calculated in the airplane's computer system from altitude, AoA, weight and loadfactor.
Unfortunately this is not that easy. As stated already, AoA is perfect at low speed but very, very sensitive at high speed when we need extreme accuracy, while having mach effects on the AoA vane as well. Calculating fails as well if your aerodynamic characteristics are not the ones you assume, e.g. due to icing of the wing. Mach effects also make it hard to calculate bach from lift to speed. So finally you are using a backup which is sensitive to exactly the same effects which may cause your pitot to fail.
LDA is a truly independent system, you may possibly even use it through the cockpit windows, so it is well protected from the environment and for example any ice blocking the light path is evidient to the crew.
On the other hand, why do we need a new system, if all it takes are old pilots...