The (Old Order) Amish are the closest precedent I can think of - very strong maintenance of tradition, internal trade and a little external but tightly constrained by their values and eschewing modern technology, and without external assistance beyond a live-and-let-live approach.
Just a note - in terms of overall economy, owning shares, UBI, etc. aren't real - they're just bits and paper. Actual wealth on this scale is in terms of goods and services consumed.
There will be a general increase of human wealth through reduction in cost of providing almost all goods and services. But that's never enough - we will also need to find ways for humans to provide value to AIs and each other.
It may be there's a large part of humanity that is purely takers, and not actually providing value (not just net-negative, but no legible value input). My prediction is that won't last more than a generation or two - it's hard to know why the AIs and productive people would reduce their consumption to keep the non-productive around.
In a world where taxis and trucks drive themselves, code writes itself etc. there's several ways humans can fit into such picture.
Perhaps they'll get rich by owning shares, perhaps UBI will be introduced, perhaps they'll move to jobs higher-in-abstraction-ladder using AI as tools, perhaps we'll just serve as testers/aligners/goal-setters.
But there's a possibility we'll have nothing to offer to such "ascended economy", or perhaps big part of a society will not be needed, while some other small part will benefit (Elyzium-style).
(I'm assuming the "progressive" part of the world is not hostile, just doesn't need the "traditional" part - in particular they respect their property rights)
Assuming this later scenario: is it possible that "traditional people" form some kind of closed system in which they still trade with each other, ignoring the outside progress?
In particular, would it require some strong coordination between them to refrain from buying/selling to the outside, or would it rather be the most natural and selfish thing for them to do try to trade locally?
Has it ever happened in the history, that a tribe has successfully walled off (without an active help from the outside to protect their customs)?