The major issue I see with LDA (obviously meanwile called LIDAR) is that you need at least two reference points (in 3D at least 3) on the aircraft. Not easy on an object which is subjected to quite some deformation. An Aircraft is not a laboraty floor. The referenze points move realtive to each other with temperature, pressurisation, air loads, acceleration, thrust setting etc. I worked with LDA 20 years ago in a wind tunnel, and even there we had issues because auf relative movement due to temperature changes and related deformation. But this is nothing which could not be solved with a few more time and money.
And I still believe it would be a truly independent system, so the overall reliability would no longer be affected by common failure causes, to add this system would reduce the probability of total failure much more, than adding another pitot would.