The very fact that such a classification has a name tells you that they are not as welcoming and accepting as they want you to think. I am nor aware of any other group of chassidim or litvaks for that matter, that have first and second class so clearly defined that there is a name for it.
I’ve heard stories from multi generational Lubavitchers, and how they are treated as second class in shidduchim, because they haven’t earned Gesze status. The message is clearly, put on our tefillin, drink our lechaims. send us your checks. But don’t dare think about marrying my child.