Moving to southern South America possible by train or by air is the only option?

The major international train route connects Buenos Aires (Argentina) with Asunción (Paraguay). There are also direct connections to Bolivia, Brasil, Uruguay and Chile, and between these countries.
However, at least here in Argentina, most of the train system was dismantled through decades of corrupt and inefficient management, and what is left of the national railways is quite poor, with very few exceptions. Train services are not only unreliable, but also quite insecure here.
I don’t know the situation in other countries, but I would not rely on train as means of transportation unless you’re looking for an “adventure-tourism” type of experience.