Yep, the reason they implemented MCAS was because they popped on some larger engines, which were too close to the ground, so they moved them up and forward on the wing a bit. This gave the plane a tendency to tilt up which could cause a stall, so Boeing added some software to bring the nose down.
In both crashes, one of the two Angle of Attack sensors was incorrect. Boeing only use one AoA sensor per flight (alternating for some bewildering reason), and in this particular flight it was constantly incorrectly stating the plane was nearing a stall. So MCAS would continually kick in until the Pilot could no longer keep the plane in the air.
> "This gave the plane a tendency to tilt up which could cause a stall"
That tendency also exists on Classic and NG 737's, and is something pilots are trained to compensate for. The issue with the MAX is the degree of this tendency changed. MCAS was meant to emulate the old (less severe) behavior.
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.
In both crashes, one of the two Angle of Attack sensors was incorrect. Boeing only use one AoA sensor per flight (alternating for some bewildering reason), and in this particular flight it was constantly incorrectly stating the plane was nearing a stall. So MCAS would continually kick in until the Pilot could no longer keep the plane in the air.