All of that ignores, of course, that the culture of mankind evolved dramatically between and during those big migrations, and then again ignoring that Africa, Asia, Europe, and America all have wildly, wildly different belief systems, cultures, politics, and people.
Who is saying that we'd take our same problems to Mars? The kind of people who were willing to move from Europe to America were not the same kind of people who wanted support the system in Europe at the time, and the kind of people who want to move to Mars aren't going to be the same people who want to continue the same system we have on Earth.
"Fix our problems here"... but what if "here" is the problem? If the Pilgrims had stayed in England and the Netherlands, they wouldn't have been able to fix what they considered to be "the problem" no matter how hard they tried. It was only by starting a new life that they could solve their problems.