Without a permanent DNI some things are very complicated, you need a lot of money which you can hardly earn from teaching English, I guess. Keep in mind that you will have to rent tourist apartments, you can not get a local credit card, but without most items in stores are more expensive. You can not get a health insurance etc...
So if you have enough money, there are no problems in Argentina. Some things are surely easier for foreigners in other latin american countries, especially getting a DNI.