Anecdotally,...
My wife's family came from southern Italy in the early '50's. Do the math. Few of them (about 50 people, in total) speak without a heavy accent to this day. Many (mainly the "housewives") still cannot hold a conversation in English despite decades of having the television tuned in to daytime talk and game shows. My recently-deceased father-in-law could not hold any kind of a conversation in English despite living the last 60 years of his life in the U.S. And he was gainfully employed from 1950 through 1985.
Many of the old guys who are barely understandable were still able to own and operate landscaping companies and make a nice life and living for themselves, putting their children through college.
I suspect that it would be more than possible for a Mexican immigrant to make a nice life for himself and his family with only a rudimentary knowledge of English. You don't have to have a "job" to achieve some success. A pickup truck and some power tools and a willingness to work hard might be enough.