My question is in statistics under what circumstances would the mode be an appropriate measure of central tendency?
Whenever you expect a "normal" Gaussian distribution and have a large number of measurements. In that case the "mode" or most frequently obtained value should also be the mean of the distribution.
It would also be best if you had a bimodal distribution — one with two modes — most likely indicating underlying two subgroups. For example, if you had a measure of handgrip strength, you would probably get a bimodal distribution, indicating different average strength between men and women.
I hope this helps a little more. Thanks for asking.