You can do the same thing with modular exponentiation, i.e. Pocklington's primality proving, but that requires finding a large factor of N-1, which is not easy

You can do the same thing with modular exponentiation, i.e. Pocklington's primality proving, but that requires finding a large factor of N-1, which is not easy in general. With elliptic curves you get as many shots at finding a group order with a large factor as you want, since if you fail the current attempt you can simply try another curve with a different order. So you can keep trying until you find a curve that has an easy to find large group order factor which, as it turns out, happens in probabilistic polynomial time.