Meat consumption *is* linked to an increased risk of cancer, diabeties, heart disease, incontinence, impotence and loads of other things. Processed meats bring moreso of a risk.
Some of them will grill, broil or roast this meat which is brings its own set of risks because of the carcinogenic affects that some of the these cooking methods bring.
In all my time of reading health articles and medical newsletters, I've not read one thing saying "eat more meat to cut the risk of ______". Usually, the gist of the statements are that a person should eat more fruits, veggies, whole grains, nuts, seeds, beans, legumes and deep leafy greens and stay away from the processed / refined foods. If a person bases their diet around this, they will be healthy but most meat-eaters won't base their diet around this - they'll stick to the meat and 3 veg method or their processed quickie foods which doesn't provide enough, IMO.
And all the meat-eaters I know eat waaaaaaaaaay more than the recommended 3oz of meat per day. Usually, it's more in the ballpark of 6-12 ounces per meal that's more the norm and believe me, more is NOT better.
Lean meats, IMO are no better than their fatty counterparts except in the only case of reduced fat. They still carry risks and they still are linked to a number of ailments. And the animals are still injected with growth hormones, bulk enhancers, anti-biotics and steroids in order for them to be ready for market in a shortened amount of time.