I can see several issues with the analysis apart from the fact that the obvious result of the study was the correlation (not link!) with obesity. This is pointed out yet glossed over in the results and conclusions:
"Role of BMI
In the present study, most of the positive associations between meat consumption and health risks were substantially attenuated after adjusting for BMI, suggesting that BMI was a strong confounder or possible mediator for many of the meat and disease associations. BMI is an important risk factor for many of the diseases examined (e.g. diabetes [7]). BMI was highest in participants who consumed meat most frequently, and some previous studies have found that high meat consumption is associated with weight gain [56, 57], but it is unclear whether this indicates any specific impact of meat or an association in these populations of high meat intakes with high total energy intakes. The associations of meat with disease risk reported here which remain after adjustment for BMI might still be due to higher adiposity, because BMI is not a perfect measure of this characteristic; we observed similar effects when adjusting for waist circumference (results not shown), but, as with BMI, waist circumference is not a perfect measure of adiposity and there could still be residual confounding."
For example cereal fibre intake was 'measured' by calculating bread and breakfast cereal (only standard types) and on that measure I would score just over zero or zero yet I have a high fibre intake - mainly from flaxseed. There is a difference between eating organic bacon and other bacon yet they are both classed as processed meat. And even t2 diabetes has a significant inheritable factor as well as triggers such as previous steroids and statin use.