If it was true, couldn't you get the same effect by taking a biopsy, fragmenting the cells, and injecting them back in? Like a vaccination, in fact. Somebody must have studied that approach already.

I assume the immune system probably already reacts to this in a specific way. For example, a major bruise has a lot of broken up cells, but doesn't warrant a big immune response.

Cancer immunotherapy is a whole field of research and treatment, yes.

Major damage tends to cause a much larger immune response than a vaccination. That said, they do have therapeutic cancer vaccines that present proteins from cancer (sometimes patient-specific) with adjuvants to help stimulate the immune response.