This is a very interesting ATL. I am considering doing a similar one about what the US constitution would be like if it a new one had been draw up after WWII. What got me interested in that was something I was reading a while back, about how apart from the massive cultural differences between them, one of the reasons why European nations were far more left-wing that the US after WWII was that they had recently drawn up constitutions, whereas the US had one from the late 18th century. So the US constitution reflected classical liberal values but the European ones reflected social democratic values.
More to the point of this ATL, it is hard to tell what would be put in a modern constitution. Much would depend on how the convention delegates were chosen. If we instead have something like a 'Founding Fathers' situation, where there is a group of non-directly elected politicians and legal experts at the convention, then I can see them possibly being the modern version of the 'Libertarian Enlightenment Deists' (not literally but the modern version thereof). If they are more demoncratically elected, then this wouldn't be the case of course.