There are some groups that did not and some of whom still do not participate in Social Security, certain state and federal government employees comprise one group.
Also, noncitizens in many cases do not have an SSN. In any case, you can decline to provide the SSN (or provide an alternate TIN or EIN number). It's basically a piece of data as opposed to an identification document. (There is the flimsy paper card, but I doubt any significant numbers of Americans have it on their person)