Well, ignoring the glaring non-sequitur that if death certificates are (as stated) the "only" way to determine cause of death, one wonders how the esteemed professor found any information to compare the death certificates to - and then concluded that the alternative source of information was the correct one, cause of death should never be determined from medical notes and interviewing the doctor responsible for the patient at the time of death, not death certificates. Death certificates are signed by anyone who happens to be on the ward who has a medical degree, so that the body can be removed for the ward and another patient in the bed before it's gone cold. Of course they are routinely wrong!
It doesn't follow from this that 1/4 of people told by a medic they have cancer don't have cancer. It is notoriously difficult to pinpoint an exact cause of death for most people. People who die of MIs within half an hour of discovery will typically be recorded as "sudden cardiac death". As for primary cancer lesions, once cancer is advanced enough to kill it can be hard work identifying the primary lesion from the others. Even on autopsy it's simply not possible to flatten a 3D corpse to 2D to find everything that's wrong with it. What about a patient with leukaemia on a haematotoxic drug which weakens his immune system causing an overwhelming bacterial endocarditis resulting in cardiac failure. Was the cause of death (1) leukaemia, (2) iatrogenic (drug-caused death), (3) bacterial infection (4) bacterial endocarditis (5) cardiac failure. You tell me. I've been discussing cause of death of patients for years and I still don't know!