Couldn't Boeing just put in a bunch of gyroscopes into the chassis and use that as a backup AoA sensor? (I am not a pilot so I have no clue if that is reasonable)
The plane attitude (what you can sense with gyros) and angle of attack are very different things. They only coincide in level flight with no wind (the attitude is measured to the earth frame of reference (transposed by gyros) and the AoA is measured with reference to the local wind stream)
Indeed, part of this story is that the MCAS system was not classified as "critial" (i.e., a system which can cause a crash if it fails). If it had been, then it would have faced additional scrutiny and they'd probably needed three AoA sensors and majority voting. Apparently the authority of the MCAS system was increased during development, with the criticality judgement based on the old smaller value.
