Yes; the problem is that for a worker being paid under the minimum wage to rectify this issue, they probably need to produce their work authorization documents at some point, and therefore risk deportation/worse.

Why not simply get work authorization documents then?

I'll assume you haven't lived in the US.

It's not easy to get work authorization documents because there is a huge queue for legal immigration (for example, my friend's cousin just moved here from a country on the Indian subcontinent, after having waited more than 15 years for a visa), while people in Mexico/Central America/wherever need to improve their situation NOW; they simply cannot wait the decade or so it would take to obtain real documents, even if they had the money required to even start the process.

Living and working in the US without authorization is a criminal offense, and if caught by someone who cares enough, illegal immigrants will be deported and permanently barred from obtaining legitimate work authorization.

Of course, there's always the option of counterfeit...