I think that may be really ambitious for a first South American trip unless you're a true road warrior (and it helps to speak Spanish too because you will not find English spoken in out-of-the-way places, at least in Argentina). I would pick one country instead of zigzagging as your itinerary suggests. Don't underestimate the distances involved (as well as renting a car in one country and driving it to another - don't know what's allowed). I drove just in northwest Argentina (Salta area) through a mix of beautiful, empty and shifting landscapes, a wine route, and small Andean towns (the largest colonial city was Salta). I drove about 1,800-2,000 miles or so in 9 days. It was not hard driving, but still grueling. There were plenty of unpaved, small roads but no cars whatsoever (I was fined after returning the rental for damages from the gravel and wind damage to one door). I am really glad I could split the driving and not have to do it all. As wonderful and fun as it was, every day was incredibly long and tiring. I would not underestimate the distances and the empty land in South America. Also, you should find out about the condition of the roads just so you know what to expect. It was a road trip of a lifetime, but a LOT of driving. I rented a manual transmission.
Check out the Footprint Guides for both countries. Like I said, I would pick one but not both. The best use of a car is not to go to large cities like BA or Santiago, but to see the smaller ones and just the vastness of the landscape. That's why my trip was a combination of a road trip in the northwest, a few days just in BA and Salta, and 2 internal flights (one from BA to Salta and another from BA to Iguazu Falls). Don't assume that the weather will be the same even in summer given the differences in the regions like Tierra del Fuego and elsewhere.