Generally that might be true, but I think if carbon black ink caused cancer with any nontrivial probability, the cancer would be likely to start where the tatto

Generally that might be true, but I think if carbon black ink caused cancer with any nontrivial probability, the cancer would be likely to start where the tattoo is placed, and people would quickly decide that tattoos were dangerous because they would see the tattoo melting into a cancerous lesion. I’ve literally never heard of skin cancer that come from a tattoo. Tattoos are so common that we would see this effect if it happened.