Looks like there is availability in SAAver First Class for these dates:
Thu April 11 JFK-EZE
Mon April 15 EZE-JFK

Beyond that, I agree, availability looks very sparse. In the past, space has been much more open. I flew AA to Rio for New Years last year, and have booked friends on trips to Buenos Aires, all direct and in SAAver First Class. Maybe American has put new limits on award seats?

Some thoughts:
If your goal is just to visit a South American city for a long weekend, have you considered Santiago, Chile? Oneworld partner LAN flies there direct from JFK, and their 2-class planes have fully-flat seats in Business. Use the BA website to check availability.

If you're flying direct to South America using AA miles, remember that NYC counts as your North American International Gateway City. This means you can make NYC a stopover and add a domestic one-way for free to both ends of your trip.

If you run out of alternatives, flying to EZE on LAN via SCL is not a bad alternative. You'll spend an extra 3-4 hours in transit each way, but the long-hauls are all red eyes so you still get to save vacation days like you mentioned.