"Accumulated capital (like Notre Dame and the Louvre) doesn't show up in GDP either, and if anything tends to lower it."
Notre Dame and the Louvre are a couple of the things that make Paris the single biggest tourist destination in the world, from which it derives a huge amount of GDP. The French tourist officials make sure to monetize the Louvre for all they can.
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"Mississippi's GDP per capita being similar to that of France, Italy or Spain is really hard to reconcile with the fact that many of us would love to live in the latter countries and couldn't be paid to move to Mississippi."
No, it isn't, because people wouldn't love to move to France/Italy/Spain, they don't even want to. People say they'd love to live in Spain/Italy/France, but the fact that they don't reveals it's mostly cheap talk.
There are some exceptions in professions that make well above the average for these countries (also retirees, students). Europe is really very nice if you make well above the average, but this is not (by definition) the typical life in those countries. Very few Americans are excited by a 24k take home salary in a mid-sized French town, or 15k in an Italian formerly wealthy industrial city.
The vast majority of migration flows are up the GDP gradient.