And of course, there's always a more sucessful MERCOSUR, or more recently UNASUR. In fact, I wouldn't be surprised if by the middle of the decade, South America achieves an EU-like level of integratiom.
I'm always surprised that this forum dismisses South American unions as ASB inmediatly without fail, yet the European Union has managed to emerge from a continent that was well known for its periodic wars and extreme nationalism before it.
You're right that's it's absolutely possible, but there are several massive obstacles.
1. Institutionalized corruption. Brazil is corrupt down to its core, and it's the biggest power, so who's going to enforce anti-corruption measures? Chile/Uruguay/other countries who get their shit together before UNASUR becomes significant won't join if this isn't dealt with.
2. Non-democracies. Obviously a UNASUR would support democracy. But the political situations in several countries definitely put a test to the definition of that word. Venezuela is obviously not, and they'll have to require Venezuela to reform if UNASUR wants any legitimacy, but what about a country like Argentina? It's in a very grey area.
3. Freedom of movement and equalization fund. It's sort of partially in place (freedom of movement) already, and already a massive disaster. You think there's a big difference in development between Poland and the UK? Try Bolivia and Chile. Or Paraguay and Argentina. Uruguay is closer to joining the EU than Paraguay is to being able to compare to Colombia. You think the Uruguayans and Chileans are going to give up their newfound riches to invest in countries that are deeply corrupt?
The EU didn't have nearly the levels of corruption, anti-democratic governments, or wealth gaps. There's a reason why the continent of war came together before the continent of relative peace, and it's all of that in combination.