direct contact with cattle is immaterial to the points that I am making.
1. Native Americans did not have exposure to the diseases that Europeans brought to America because Native Americans did not have domesticated animals other than dogs.
2. Europeans brought some diseases to the Americas with them because of having 3 or 4 millennia of experience with domesticated animals who were vectors for diseases to humans. Exposure to the diseases from those animals had given Europeans (and some Africans) stronger ability to cope with those diseases, or even to acquire immunity to them.
3. It was the animals who made the difference between exposed and unexposed populations to diseases that hit Native Americans (and Hawaiians and other virgin populations) so hard.
4. Without prior exposure to cattle, like Europeans had had, Native people on the Plains were vulnerable to severe infection and death. The people who distributed those blankets knew of the Native vulnerability which is why they did it.
5. At the time of the encounter of Sapiens with Neanderthal, domesticated animals were not a factor like they were in the encounter of Native Americans and Europeans. So if Sapiens carried diseases from Africa to Neanderthal groups in Europe and Asia, there had to be a means of carrying the diseases other than by introducing African animals to Europe, like what happened when Europeans introduced their animals to the Americas.
6. With your expertise in epidemiology, perhaps you have some good knowledge of, or insights into how Sapiens would have carried African diseases into Europe and Asia. Wouldn't Sapiens have been exposed to new diseases, too, in their new environmental exposure to conditions and species outside of Africa?
7. Some speculation on genetic inheritance from our mixed ancestry. Since today's Sapiens carry Neanderthal genes and mutations that benefit us in some disease exposures, it looks to me like we benefited from intermating with Neanderthals. Perhaps they did not benefit as much from intermating with us. The hybrid offspring of the intermating who inherited the best combination of genes and mutations for coping with diseases outside of Africa survived. Perhaps Sapiens and Neanderthal hybrid descendants of those matings were able to survive better and produce enough offspring to outnumber the "pure" Neanderthals, who then disappeared except for the genes they left behind in us. At the same time, "pure" Sapiens outside of Africa also disappeared due to intermating with Neanderthals.