Weren't the Picards Amish? Or were they just Luddites?
Luddites.
With the Amish, it's not a rejection of technology for technology's sake. It's a rejection of all things they deem "English."
I live just outside the largest Amish community in the United States, so this piqued my interest. Amish are rather hypocritical about their rejection of technology. If the wife, for example, isn't Amish, then it's OK to have electricity and modern conveniences in the home for the benefit of everyone. Many Amish around here have computers, smartphones and modern conveniences in their businesses. Some have recently started to secretly wire their homes for electricity, but ask the contractors to bury the lines so the neighbors don't notice.
Amish, or Simple Folk, thrive financially on tourism and selling hand made goods and furniture. We go to Amish Country about 3 or 4 times a year to buy food, particularly
cheese and
candy.
There are Amish colonies in South America. My step-son visited one when he was in Bolivia. The Amish aren't about proselytizing, but are about living separate and independent of mainstream society. They want to be left alone to mind their own business. At the same time, they want outsiders to come and buy their goods and services.
I could see 23rd or 24th century Amish scattered about on colony worlds, especially on worlds similar to Tyree's planet or maybe Capella. Not necessarily those worlds particularly, but worlds with a primitive (by Federation standards) technology level that would enjoy trading food and goods with Federation citizens. Worlds that would leave them alone yet appreciate their services.
The colony from DS9's "Paradise" could have been Amish with a few minor adjustments. Of course, the episode then wouldn't work because the entire colony would have rejected technology by choice, not out of trickery.
