Trading the goats for chickens doesn’t necessarily lead to more wealth though, that would depend on using the chicken or goats to improve productivity.

Next door to where I was recently staying I talked to some farmers that grow rice. They live a subsistence life, barely earning enough to survive by selling their rice. Trading their rice (equivalent to the goats or chickens in your example) does not make them wealthier. No matter how much rice they trade they aren’t going to get a bigger slice of the pie or make the pie bigger. They can’t grow other crops without investment in education, seeds, and technology, which they can’t afford.

So in that kind of real world scenario you cannot simply trade what you have or produce to increase your wealth. And for most people they don’t even own the chickens or goats or whatever. They don’t even own the land they live on (land is absolutely zero-sum). Increasing their wealth requires outside investment of education or technology, which isn’t trade but charity/redistribution. And not surprisingly, the way that has happened has been through government investment (redistributing the pie).

The Robinson-Crusue Econ 101 hypotheticals are divorced from the reality of economic development.