Yes, all true, but does that mean anything for the person whose most intimate contact with the animal is picking up a chunk of its corpse packaged in plastic in a grocery store, cooking it, and eating it? Granted, eating its body is pretty intimate contact IMO, but how many of those pathogens are actually passed by eating meat? I think they originally crossed the species boundary through the people who were in close contact with them while they were alive, yes?
I believe you are quite correct, most of the risk of contagious disease acquistion comes from caring for animals, slaughtering them and butchering them, rather than eating body parts that have been heated to a high enough temperature to kill pathogens. Still, you can get some contagious diseases by simply handling uncooked body parts. Even carefully cleaned corpses will have traces of fecal contamination. Even when being normally careful not to use, for example, the same cutting board for cutting up animal parts, as for raw vegetables, and when being careful to wash hands after handling animal parts, and before handling vegetables, contamination of vegetables does occur. In actual practice, few people are really scrupulous about handling vegetables after touching corpses. They forget to wash their hands inbetween, or go back and forth between cutting up pieces of meat and cutting up pieces of vegetable so many times within the course of just a few minutes, that if they scrubbed their hands every time they technically should, to avoid contamination, their hands would be raw by then.
But yes, most of the problem is from sleeping near by to animals, walking thru barns, touching things they touched, and keeping them alive when they are sick, rather than getting rid of them as soon as they show any sign of contagious illness -- because they are expensive and wastefulness is associated with fear of not having enough to eat.
Simply by living in a society where animal husbandry is prevalent, and raw animal flesh is passed around, we all risk getting certain contagious diseases, either directly from the animals or from people who got the disease from the animals. Simply picking up and swallowing a piece of heat-treated flesh adds little or no extra risk.