I think the logical fallacy is thinking about the economy as a steady state system. At any given time there are finite resources, so it is "zero sum" in that the pie is finite and slices can only be shared once.
If I have 10 chickens and you have 10 goats, we can exchange between us and each of us would simultaneously benefit more from having 5 chickens and 5 goats each, even though the pie hasn't changed.
Or maybe I prefer to have 8 chickens and 2 goats, while you prefer to have 8 goats and 2 chickens, and both be happier as a result by exchanging between us while the entire pie has remained exactly the same.
So happiness/usefulness/value was created by the exchange of goods and services even though the pie hasn't changed at all.
Also, these preferences can change over time, so value can continue to be created even if the pie doesn't change.