Are you planning on touring south america by car? If you are going to travel for more than 2 months, renting is probably not the ideal option for you.
Renting is Expensive
A rented car costs around 100$ per day, which turns to 700$ per week which turns to 3000$ per month etc... while when buying a good car using Suzi Santiago's service you are most likely to lose around 10% of the car's value +- depending on the time of travel and KM added but the average traveler loses around 1000$ for about 6 months of owning a vehicle instead of renting it (which would cost around 18000$ for 6 month).
Most rental companies will find reasons to even charge you more with their penalties for visual damage you may have added to the car (always happens when traveling), however when selling your own vehicle to other travelers, most of them wouldn't care about a bump here and there as long as the car works great. And even if you do want to fix some visual damage of the car, it could be done for cheap at workshops we know and trust.
Can't Turn it to a Camper
One of the biggest joys and necessities of overlanders is to modify their cars, turning them into campers, weather its an SUV or a van, adding a bed or kitchen, maybe some solar panels, isolation, parquet flooring, even a shower and toilet. These features can be crucial while traveling for a long time, reaching to deserted places, camping in the wild but still enjoying the comfort of a mobile home. All of these, can't be done in a rented vehicle obviously, however, they will even raise the value of a self bought vehicle being sold to other travelers.
It's Not Yours
Overlanders always fall in love with their vehicles and become attached to them. A rented vehicle is simply not yours and doesn't feel like that.