That might be addressed by modifying the animal to produce the dsired organ at the correct size, scrubbing out all livestock cells with strong detergents, then seeding the organ extracellular matrix with progenitor cells from the intended recipient.
There would be no livestock cells left, so no worry about zoonotic pathogens.
The only known mammalian prion is PrP, which noticeably affects animal behavior, and is detectable by testing lymph. If the animal has prion-related encephalopathy, you burn it.
Viruses cannot reproduce without living cells. Viruses that employ lipid capsids would likely be removed by the detergent along with the cells. Some viruses do embed in a type of extracellular matrix (HTLV-1). As far as I am aware, no research has been done on the transmissibility of viruses in decellularized animal organs from donor to recipient.
I presume that those facing imminent organ failure would rather catch a cold from a pig and recover in quarantine observation than die while waiting for such research to conclude.
There would be no livestock cells left, so no worry about zoonotic pathogens.