The ape diet
Here is a trial case (it does not really meet the standards needed to be called an "experiment") in which nine human volunteers were put on an ape-like diet for 12 days. In some ways the diet was adjusted for human tastes; insects were not on the menu, but fish, which wild apes do not eat, were available. Root vegetables, which wild apes also do not eat but for which our hominid ancestors acquired a taste long ago, were also included. Nevertheless, the diet did broadly resemble the way apes naturally eat. Despite the short duration of the test, the health benefits were clear.
To respond to one point raised in the comments: cannibalism does exist among chimpanzees, but it is extremely rare. More commonly, chimpanzees hunt other animals for meat, but they still consume far less meat than most modern humans, partly because they are not very successful at hunting. As far as observers can tell, the other three great ape species (bonobos, gorillas, and orangutans) do not eat meat at all.