I'd have thought that sticking to a diet for 20 years, and being very fit and active at the age of 67, would be pretty compelling proof that there aren't that many dangers being stored up! 😎
As ever though, this is lazy reporting. The Atkins and Paleo diets are NOT high protein diets, they are high fat diets. You'll also notice that the people studied had over half of their calories come from carbohydrates, so clearly the people studied were not on low-carb diets or any of the branded diets accused of being a problem. If I were a more disruptive person, I'd suggest the average diet on test was actually far closer to that recommended by Diabetes UK/NHS etc., rather than a low-carb diet!
In fact, the ones on 'moderate' protein were typically eating the level you'd expect - Atkins/Paleo is generally 70% fat, 20% protein and 10% carbohydrate so even their protein consumption would be 'moderate' by the standards of the sample.
Furthermore, there is little distinction about the kind of animal protein. My gut instinct is that eating a lot of beef from cows raised on corn, hormones, antibiotics and god knows what else (as would be the case as this was an American study) carries a greater risk of danger than eating a lot of beef from more traditionally raised cows, which may be the case in Europe.
A four-fold increase in risk is also not statistically interesting without a baseline to go on. It might sound like a lot, but if your risk of heart disease on a plant protein diet is 1%, a four-fold increase is still just 4% - not an increase that's likely to discourage me from eating cheese.
So in other words, potentially a lot of nonsense in this study.