Family friend has been renting an apartment for years in Greece and him and his wife are able to do most of their work remotely. It's in a desirable area and if a tourist inquired about it, they'd probably pay about 1-2k/mo. However, because he has EU citizenship and speaks Greek, he's paying under 5k/year for the place and even got them to include furnishings. They've spent as little as 2 months there and as much as 8 months straight.

If you don't already have them, you may want to make some local friends in the area you love and see if they can help you find a place within the community instead of going through a realtor that knows you're a foreigner and presumes you are wealthy.

Many countries in EU don't seem to care if you rent vs own for just a permanent residency permit. Some countries will fast track actual citizenship if you DO own, but that may not be all that beneficial if you can get a permanent residency application through. You also have to consider the pain in the ass of taxes and other rules between the US an whatever country in the EU that you are talking about.