The way humanity keeps its DNA clear of cancer over the long term is because people with cancer die and healthy people don't. A slow evolutionary dynamic keeping a fast evolutionary dynamic in check. This shouldn't be controversial.

The mechanics of it, is like I said, the population bottleneck. Cancer cells are parasites and the bottleneck leaves them without a host.

A fascinating cancer is "Devil facial tumour disease". It has learned to spread from animal to animal, thus bypassing this purifying mechanism. But as it no longer consists of the animals' own cells it may be argued that it has become its own species and should no longer be considered a cancer. Supporting the speciation argument is the fact that the cancer cells even have a different amount of chromosomes from the animals.