Regarding radio-altimeter not working on cruise altitudes, I presume it is because of the time it needs for the beam to go and return back from earth to the airplane which has already "gone" forward?
well, it is about the much higher power you need to receive back an echo from 30.000 ft away, and you will have too much scatter from all the different hills on the ground reflecting.
LDA is a great ide, but it works only because there are particles in the air. I am afraid it is too clean up there, to really work. Otherwise, it is absolutely perfect, as it is a truly independent source of information.
AFAIK, AF447 had altitude read-outs at all times- it was airspeed that was in error.
The mach correction of static is done by computers using airspeed info, so at least the altitude information was "uncorrected", but probably much more exact than a lot of other information.