Its not the rawness of the meat that is the issue, both humans and [omnivore-carnivore] animals can digest raw meat just fine. Biologically our guts are perfectly adapted to digesting raw meat. Its the food-borne illnesses on the raw meat that are bad for us. Both humans and animals are susceptible to contracting food-borne illnesses (e.g. bacteria, fungi, toxins, parasites, worms) from eating raw food. The difference is that humans cook their food which reduces the risk of contracting a food-borne illness. Since animals cannot cook their food they are more susceptible. Some animals, like vultures, have adapted digestive systems that are better able to prevent food-borne illnesses they but these are not 100% effective. Therefore, in regards to contracting a food-borne illness animals would benefit from a cooked food diet if they had the chance. Humans that decide to eat raw food (e.g. steak tartar, raw sushi) are taking a greater risk, but if they know that their raw food has been properly sourced and prepared they can enjoy it will minimal risk to themselves.
A few additional points:
Animals also get sick from eating contaminated food and water sources. To be clear both humans and animals can get a waterborne illness from drinking contaminated water, and foodborne illnesses from eating contaminated food. This is true for carnivores, omnivores, and herbivores. Some species are better equipped to handle foodborne and waterborne illnesses, but there is also considerable individual variation. This includes people & animals who are living near the same untreated water source their entire lives. For example, you andyour pet dog can get giardia from drinking untreated water from rivers/lakes.
Generally, animals and humans can withstand foodborne illnesses to an extent without compromising health but it is difficult for the old, young, weak, or sick to withstand these events or episodes. Wild animals and humans die from contracting food borne or waterborne illnesses. It is the increased risk of contracting a food borne or waterborne illness that is the problem with raw food or untreated water. Generally speaking, the vast majority of wild animals will have a nearly continuous parasitic load, many of which are contracted through eating contaminated food sources or drinking contaminated water. More on Parasites and Foodborne Illnesses by USDA.
Foodborne and waterborne illnesses are typically caused by a bacteria (e.g. E. coli), a parasite (e.g. tapeworm), fungi, or toxin. For example, toxins can be found in some mushrooms, in shellfish (red tide), or the Fugu fish (puffer fish) toxin. Both house cats and dogs can get salmonella poisoning. I am going to assume that large cats can also contract salmonella poisoning, although risk may vary between populations (e.g. wild large cats vs. captive large cats).
ALL raw food has the potential to make you sick, including veggies, fruit, fish, and meat. For example, E. coli outbreaks on veggies like tomatoes and spinach are fairly common. Foodborne botulism is caused by a toxin produced by a bacteria that is often found in improperly prepared canned/preserved foods. Raw fish in sushi might also contain parasitic tapeworms - which you can contract.
Cooking food includes many processes such as boiling, baking, peeling, washing, freezing, or grilling. These process kill or get rid of potential sources of contamination. Animals would also benefit from cooked food diet if they could cook. The same applies to boiling water to kill pathogens within it...animals would also benefit in this regard.
While our modern food industry takes every precaution to prevent a foodborne illness from getting to you, its not 100% perfect. Some raw foods have a higher risk (like raw chicken) and cooking using proper and safe techniques is highly recommended to reduce risk to yourself & your pets/animals in your care.