"the altered DNA is also incorporated into our own DNA when eating genetically modified foods"

The article doesn't imply that entire genes (modified or not) are incorporated into our own chromosomes from eating such, only that heat denatured nucleosides (the coding elements of genes, after phosphorylation) can be. Genes would generally be broken up by nuclease enzymes in the small intestine.

Anyway, even though I would stay clear of GM food it's obvious that cooked foods and meats cannot be the cause of modern cancer rates. Diet might play a role, but the effect is not driven by consumption of cooked meats.

Why do I say this? Because colon cancer incidence has tripled among Americans since the early 20th century, when people in the USA ate a lot of meat (and cooked meat too, raw foodism is a recent trend).

Per Roger Horowitz in 1909 urban US Americans in the lowest income brackets ate ~136 pounds of meat per year, the richest more than 200 lbs. During WWII the meat ration for US citizens was reduced to 28 ounces per week, or 1.75 pounds, which is still quite substantial.