I don't think they'd have needed a model for why it happens to see the obvious correlation between using things used by a sick person and getting sick. It'd be compatible with the older idea of a "miasma" spreading the disease.
I agree. Evolutionary speaking, animals that develop different behaviors based on 'clean' versus 'unclean' would have a leg-up against the competition. Ants, to give a quick example, throw their dead out with the rest of the garbage.[1] (Which can lead to a funny experiment to have them carry out living members as well - "I'm not dead yet!"[2])
You don't have to understand why you are doing something - or even that you are doing it at all - as long as it works and gives you a benefit, however small. Without proper understanding you will also have false positives, but it's better to be assume there is danger than to assume there isn't any. False positives outweigh false negatives. (To a certain level, of course, too much of it and paranoia becomes harmful again.)
they don't necessarily need to figure out the germ theory to know that diseases spread from proximity and/or contact with sick people when it happens right in front of them.
Also, there could be many such reasons for burying/burning someone dead with his/her possessions. My point was that almost always it becomes "elaborate rituals and beliefs of afterlife with same worldly possessions that they had here on earth" - just because we know pharaohs/ancient Egyptians did it.