It isn’t known how many of the total immigrant population arrived legally vs. crossed the border without permission. What we currently know as fact is based on relatively recent increased tracking of visa overstays. By estimates, the number for the total population shouldn’t be all that lopsided toward illegal entry over overstayed visa. Are you trying to argue that the number is 0? Because otherwise any argument you’re making is totally irrelevant anyway.
For the question of a term becoming considered to be an offensive slur it wouldn’t matter if it was 2%. It still could become an offensive slur. Respecting the sensibilities of a minority class pretty obviously can’t be based on them being a majority.
You still can’t assume that any one unauthorized immigrant is a criminal based on the idea that at least some of them must be, or that just because they are working they necessarily falsified any documents. Do I understand correctly that you are or were a criminal defense lawyer? It seems very odd to need to explain this to you in that case.
Besides coming with their own savings or finding a bag of money on the ground, there are a whole lot of other possibilities that don’t involve fraudulent papers.
Maybe someone is paying them to do day jobs without doing any identity check or reporting. Maybe they are self-employed. Maybe they live with family or friends who are legal residents or citizens. Maybe they collect cans and turn them in to recycling centers.
You’re already swimming upstream by arguing in favor of a term that is well on its way out of polite use regardless of any arguments against it. But the argument that “Some of them must be doing something illegal so it’s accurate to call them all illegal.” is still remarkably stupid.