Literally the only thing they seem to say for sure in that paper is that they expect to see bone evidence they don't see. While I don't doubt the earnestness of the paper's authors, their primary study seems to be in Egyptology first. They practically gush about how the ancient Egyptian medicines are still in use today...but then only list ones based on papyrus. They don't account for the fact that our sedentary lifestyle is shown to increase cancer rates, they barely explore different cultures, with only a passing reference to Greece, who has numerous treatments, including excision to remove cancer, and admit that the ancient Egyptians didn't even classify benign tumors differently than no-benign. They don't account for the increased consumption of meat in a western diet. They don't account for massively higher numbers of humans.

Based on their paper alone I can think of at least 5 other hypothesis that they didn't even cover.

1. Periods of fasting and lack of food decrease cancer. 2. Less sedentary lifestyles decrease cancer risks(this has actually been shown to have some promise, with HIIT workouts showing improved response rates and decreased reccurance) 3. Decreased infant mortality caused an increase in susceptibility to cancer because weaker physical specimens are surviving. 4. Firearms. Cancer rates have increased with the output of firearms. 5. Decreased prayer to the Egyptian gods. As these gods declined cancer rates increased.

Some of these are obviously less serious than others, however it doesn't excuse the plausible possibilities. These aren't oncologists, their paleontologists, also known as historians who guess about things based on what they dig out of the ground.