I'm a general-, not a micro-, biologist Mike, so I cannot answer that definitively; but I doubt "almost every single disease" is animal-originated. I suspect Bill may be over-emphasising a theory about the origin of HIV from FIV and SIV and then lumping in the bacterial and fungal diseases that are a LOT more easily transferred across the species barrier than viruses.
We're animals too and we have a lot of human-specific viral diseases that are not found in other members of Animalia. I'd further remind you a lot of viral diseases are the result of 'natural' mutation and subsequent selection, that can transform them from base virus (relatively unimportant, but in a 'reservoir' somewhere) into a form that one (or more) species may be then be highly susceptible to. This can mean, harmless to humans before mutation/selection and harmful afterwards.
My cross-species comment originates in the selection necessary for a virus to thrive in one species; its genetic sequencing must be specific to that species, adapted to overcome ITS physiology/cytology and would not normally be viable in a different species with differing physiology and crucially, immunology. There ARE viruses that buck that rule, just ask any veterinary surgeon; but they are largely 'irritations' and controllable, not epidemic originators.
Sam