I don't believe that our paleolithic ancestors ate a lot of red meat. At least compared to how the "paleo" community is doing it. Our paleolithic ancestors probably ate collagen and organ meats instead.
I will speculate that our paleolithic ancestors got their majority of their calories from eating root vegetables, and not from eating animals. Yes, this is contrary to what the "paleo diet" envisions.
So what justifies this theory?
The first reason is that our ancesters ate a plant-based diet four million years ago. We have hunted animals only recently, only after we have invented tools such as the handaxe and spears.
The second reason is that meat in high in iron, and that humans don't have a mechanism to remove excess iron. Thus, this means that humans aren't evolved to eat large amounts of meat.
The third reason is that some of our ancestors have larger snouts. Large snouts persisted even to two million years ago. Yes, even archaic Homo sapiens, who lived 100,000 years ago, have larger snouts than us. As you know, large snouts are useful in eating plant foods. Large snouts help us to chew raw sweet potatoes and other root vegetables more easily.
There's a theory that we evolved smaller snouts became more carnivorous, thus eliminating the need for large snouts to chew plant food. But I don't this theory is correct.
Even if our snouts receded in size, this still isn't evidence that we ate more meat. Actually, cooking may have caused this. The invention of cooking made sweet potatoes softer to chew, because heating them breaks down the cell walls. So we don't need large snouts to eat cooked root vegetables. This may be the reason that our snouts evolved smaller.
The fourth reason is that our ancestors probably lived in the savanna, where the root vegetables are high in Calories. This encouraged our ancestors to dig up root vegetables because they're "fallback foods".
The sixth reason is that a starch-based diet is more muscle-building than an animal-based diet. Starch refuels glycogen, and it can be directly used for energy. It's common for low-fat dieters to experience lowered athletic endurance. This means that humans do best on a plant-based, high-starch diet. We do best on a plant-based, high-starch diet because we are evolved to eat a high-starch diet. If it's the other way around, then why will we experience poorer athletic performance on an animal-based diet? So there's must be a reason that we do better on a high-starch diet: we are more adapted to it.