I use the round the world fare about three times a year. For around the world it is a great savings. I do not think your itinerary qualifies since you are only going to South America. To qualify for round the world you need:
1. Minimum of 10 days
2. Three international stops
3. You must go around the world. Example of my itinerary: New York- London (UA); London-Franfurt- Johannebsurgh ( LH): JNB-Singapore (SQ); Singapore to Sydney (SQ); Sydney- SFO (UA); SFO-JFK (UA). This starts at about $3,000 for economy $5,000 for Business and $7,000 for First.
You are not going around the world. You might want to look at a VARIG Pass in South America. Take United to your first detination and look into a pass.
The round the world in First is about the same as flyng JFK-London in Business, so it is a good deal.