Middle class people from wealthy nations who move to poor countries are relatively rich by the standards of their new country. Middle class people from poor countries who move to rich countries are relatively poor by the standards of their new country.
If you are middle class in EU and have little savings, moving to US will do the opposite and you would be better off.
Your comparison holds for the inverse: getting most of your quality of life from static wealth.
I didn't say it'd be easy. I'm just pointing out that only one of the two groups being discussed has a real choice in the matter.