"Earth’s Roche limit, the distance from the Earth where its tides could break another object apart"
It's a very common error. This limit is for an object "held together by gravity alone"! The moon is not held together by gravity alone - the dust on the surface is, but the rocks in the center are all fused together.
Additionally since they assume the moon will continue to have a synchronized orbit (the same side always faces us) it won't feel the tides in the same way. It would feel some constant stretching, but not the severe flexing caused by rotation.
The closer side would fall "down" and the far side would fall "up". Only the center would be in free fall. I suspect the moon could survive that - but I'm not sure.
If not, it would spread out into rings.
There's other errors too: In a situation like this the Earth would also synchronize it's orbit. A day would speed up and the moon pushed higher until the two matched. This would stop the flood, burn scenario in the article.