If you are interested about doing this on UA, I've flown this route twice.

The best connection is ORD-GRU-GIG and back. On the trip to GIG (UA845 connecting to UA861), It is a 40 minute connection at GRU.

The connection at GRU is virtually guaranteed. They held UA861 in GRU for us on one trip for about an hour. I talked about it with one of the cabin crew and they indicated they typically wait for UA845.

Your travel time will be about 12 hours to GIG and 13 hours on the return.

On Mileage Plus, you would be Silver after the third trip and would get free EconomyPlus (at check-in), if it is available. From my experience, E+ was pretty empty flying midweek.

After the fifth trip, you would be Gold and be able to select an E+ seat at time of purchase for free.