I guess it is my turn to be that guy.

None of those graphs should be line graphs. There is no continuous X variable.

The lines make it easier for the eye to track from one point to the next of the same series.

They also make it easy to visually compare different wear patterns for different species, especially as the teeth are sorted back-to-front. For instance, on the mandible chart, the lines make it very easy to see that the teal, red, and blue lines have similar patterns.

I agree that a stacked bar graph would be nice, since the focus is on the per-species total across all 8 tooth types.

Could you explain why a line graph is bad for a discrete but ordered variable, like [Molars - Premolars - Canines - Incisors]?

In this case a line graph is bad because the line doesn't mean anything. It implies an extrapolation that doesn't actually exist.