> Most healthy, active people who eat decently, get enough rest, and avoid drinking and smoking, will be able to eliminate cancer as it comes up
Incorrect.
There are tons of cancers that hide and mask with symptoms common to other symptoms. Kidney cancer, for example, presents pretty similarly to both kidney stones and UTIs. Even blood in the urine isn't proof positive that anything is wrong beyond either of those conditions. And, by the time blood is in the urine, it's often too late.
Liver cancer is even worse. The first symptoms you get can be thought of as a simple pulled muscle, just a little ache in the back. By the time you have appreciable problems, like turning yellow, it's quite advanced and too late to really do much.
There are common cancers like colon, skin, breast, and prostate that more fit your description of being mostly harmless so long as you get regular screenings and eat healthy. But, for every part of the body, a cancer can form and the symptoms are very often invisible.
I'm unfortunately all too familiar with how cancer looks. My wife currently has stage 4 cancer that started as kidney cancer. She does not drink or smoke, gets enough rest, and is very active.