Not sure about higher chance, even if you do have more top universities. But a big point of mandatory education is to encurage the lower classes to get a decent education and also to ensure that the youths gets at least some exposure to science instead of a pure creationist religious education. So that suggestion would further drop the US social mobillity.
William Jefferson Blythe III (later known as Bill Clinton), Al Gore Jr (aka II), George Bush II, John McCain III, Obama II.
Naming conventions tells of elective monarchy...
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