I think it works in the US because those airlines are often going head to head on the same routes, giving the customer a choice between which to use.
In South America, there is very little competition on most routes. Latam dominates obviously, but hardly even tries to compete on some other quite major routes. eg. Sao Paulo - Buenoes Aires (2 flights per day), Rio de Janeiro - Buenos Aires (1 flight per day). Latam has basically just given these routes to Aerolineas Argentinas & Gol. So if that is your most regular flight, then Latam is not the airline you want status with.
But Latam dominates the continent for most destinations & you don't really have the "choice" of using another alliance. Whilst you can find a different airline on many routes, you can't find a different airline that is the same alliance for most. eg. The two biggest competitors to Latam are Gol & Azul, both of whom are Brazilian based & focused & not a part of any alliance. Avianca is OK if you're Colombia based but if you want to fly anywhere other than the major city in Chile, Argentina, Peru, Brazil (SAO + Rio)etc, you'll need to fly or connect with someone else.
So my point is, Latam don't need to offer someone a "challenge". Either the person is flying in South America (in which case they will have one Latam flight) & they will status match them, or they're not flying in South America & they won't. The number who ARE flying in South America that don't have a single Latam flight would be very small I guess.