> Environmental influences are typically mediated by their ability to cause or promote mutation

That may be typical, in that there are definitely lots of environmental influences that work that way, but its been established for at least on the order of a couple of decades that there are also influences that impact the emergence and development of cancer when potentially cancer-causing mutations exist, and that genetics while important to cancer, aren't the whole story and mechanism. Attributes of the extracellular matrix, which can definitely be affected by physical and chemical effects that are not mutagenic, are one of the established influences.