On the other hand, if a market for a given skill has significantly more supply than demand, then the producers of this type of skilled labour may find themselves unable to sell their labour for much more than unskilled labour, regardless of how much effort it took to obtain the skills in question.

See the many creative and performing professions, where labor costs are often comparable to working retail, as examples. Those have more of a lottery flavor to them than unskilled labor does, where the best artists/musicians/actors/athletes can make much more than the median, but the median is often low. (If you've ever seen a sign lamenting that teachers don't get paid as much as professional athletes, go to the BLS website and compare their median salaries.)