Cancer is virtually unknown until people start living long enough to develop cancers. Oh and don't forget the juvenile cancers.
70 is easily a cancer age. Going by the current research at https://scienceblog.cancerresearchuk.org/2015/02/04/why-are-... one can expect 3/4 people over 60 to have cancer. And it's increasing.
So yeah the not living long enough doesn't really hold water. Also juvenile cancers weren't really a thing at all until the modern world (more likely bacterial infections until we discovered penicillin were more likely for juvenile deaths).