Your ogic was that "undocumented" didnt make sense because while they lacked some documents (greencard/citizenship) they had other documents (license, census etc).
By your own logic calling them "illegal" therefore is equally inappropriate since there are many ways in which they are legal (for example if they have a license they can legally drive) and are illegal in other ways.
so by your own logic "illegal" makes just as little sense.
by that logic you are now deunking your statements about "undocumented" and showing why it is actually a perfectly fine term. You cant have it both ways.