Even if it's not, "it's a privilege, not a right" is a bit of a platitude. "Due process" also means fair treatment. That something is a governmental license does not mean that there is no due process protection, just somewhat less and based on differing standards.

Besides, "pursuit of happiness" is not a constitutional right. It comes from that anti-government document written by revolutionaries based on the teachings of liberal radicals – the Declaration of Independence. Never made it way into the Constitution.