I think English "does not have a leg" would only ever be used in the idiom "doesn't have a leg to stand on", which can only be used metaphorically. If used literally, the listener would just be confused about whether one or two legs are missing.
However, the same construction could be used literally for any body part expected to occur in the singular: "The old man doesn't have a nose/mouth/colon", etc.
The ravages of time sucks.
The old would refer to old people generally, at least in British English, so we could say the old have no legs, but that would mean all old people have no legs. The old on its own can't mean the old man, at least to my mind. In languages where there are masculine and feminine adjectives it would make perfect sense, but to me in English "the old" in English doesn't mean anything without more information
In America you must specify the pronoun "man". If you said "the old" it would just sound weird and people would be like, "the old.. WHAT? What is old? Who are you talking about?" I don't know what country you are coming from, but in the U.S. "man" is NOT implicit when saying "the old..."