There is a limit to how many teeth can be replaced. Elephants for one, have a sliding system where the molars are replaced from the back to replace the worn ones. This only happens 5 times so older elephant will eventually die of starvation if not earlier from other causes. Sharks also only have a certain amount of replacements but they have more replacements than humans because that is required in the way they attack prey.
Most mammals have a juvenile set of teeth which is replaced as they grow up. A very young mammal with 'adult' sized teeth would not be able to eat, so the permanent pegs only appear when the body is larger. Humans already experience the onset of replacement molars later on in life with the appearance of the so called wisdom tooth. Unfortunately, they are rarely functional and not always develop properly.
Don't forget that the human body is only designed to last 40 odd years and without eating processed foods.