One reason for not eating potato skins is they can be toxic: When
you're served a potato in a restaurant you have no idea how old it
was, how green it was, or how hot it got during cooking which, note,
only partially destroys the toxic compound.

Wikipedia:

Potatoes contain glycoalkaloids, toxic compounds, of which the most
prevalent are solanine and chaconine. Cooking at high temperatures
(over 170 °C or 340 °F) partly destroys these. The concentration of
glycoalkaloid in wild potatoes suffices to produce toxic effects in
humans. Glycoalkaloids occur in the greatest concentrations just
underneath the skin of the tuber, and they increase with age and
exposure to light. Glycoalkaloids may cause headaches, diarrhea,
cramps and in severe cases coma and death; however, poisoning from
potatoes occurs very rarely. Light exposure also causes greening, thus
giving a visual clue as to areas of the tuber that may have become
more toxic; however, this does not provide a definitive guide, as
greening and glycoalkaloid accumulation can occur independently of
each other. Some varieties of potato contain greater glycoalkaloid
concentrations than others; breeders developing new varieties test for
this, and sometimes have to discard an otherwise promising cultivar.