No, they do not have the potential. They have chronic profound cognitive disability. And yes, animals have the capacity for rational thought. Have you not seen corvids problem solving? Google dolphin intelligence. There's a dolphin in SeaWorld whose tank partner died, so it took over the partner's tricks. It had never been taught them, it had only seen its partner doing them. There are adults walking around holding down a job who can't do that. You underestimate the abilities of some animals to think. Sure the average human is way better at rational thinking than the average dolphin, chimpanzee or parrot, but there's overlap. As a definitive marker of personhood, it doesn't work, even when what potential there is is actualised.
What you are trying to say is that the average human has greater capacity for rational thought than any other animal, therefore they should be classified as persons. Why? The cheetah has a greater capacity for fast running than any other animal. The African elephant has greater mass than any land mammal. The common garden snail is the fastest snail species. Why is rational thought so important? I'm not saying it isn't, I just want you to spell it out, if you think that this line of argument is worth pursuing. At the end of the day, you don't like my criteria and I don't like yours. Opinions differ. Now what?