Antivirals only prevent replication, too. Virus capsids don’t respire and have to be physically destroyed to neutralise them. Which is hard when you’re trying to keep the delicate organ around them alive. Maybe you could use magical nanobots, but then why do you need an animal organ anyway?

The actual research is more centered around stopping it from happening in the first place, and then screening for any contamination. The critters might be raised in completely sterile conditions, and have already been genetically modified to remove any included ancient viral sequences in published work. Of course, all the producers have to actually follow the regimen everywhere in the world…