The interesting thing going against your fear is that there are organisms like blue whales that do not die of cancer, or at least for them it is an exceedingly rare disease. The reasons for this are not yet understood, and they may not be entirely applicable to humans (e.g. the sheer size of a blue whale may play an important role, since cancer would take much, much longer to grow to any noticeable proportion of the whale than in a human-sized organism), but at least it proves it's not completely impossible to have life without cancer.
I can't actually find a good source for blue whales not dying of cancer. I do see lots of sources claiming the rates are way lower than in humans, and that is indeed interesting. It's evidence that we could do better at suppressing cancer, but I'm very convinced that cells that can evolve eventually deciding to go rouge isn't something that be precisely stopped. Just made less likely
I'm sure there are many side effects that could only be found and fixed on a mountain of dead babies. For example, if I understand correctly, this makes elephants very susceptible to radiation sickness, as their cells are on a hair trigger to destroy themselves at any sign of genetic damage.
