One would not need off shore accounts for money transfers to Mexico. All one would need is a person or entity in Canada, the U.K. or some other third nation to act as the middle man.
An illegal alien in Houston would wire money to a company in Canada, that company would take a cut, and they would then wire the money to Mexico.
You’re trying way to hard to defend the premise. Fact is the average day labor Illegal isn’t going to go through all that. They’ll go with the cheapest method, if it saves 50 cents after “wall fees” to buy a USPS money order, that’s what they’ll do.