Yup, certainly any bits of human flesh found count as 'BioHazards', and I rather expect that the Coroner's van does have at least one sticker inside.
As an ex medical specimen courier, I can certainly tell you our vehicles didn't have warnings on the outside, but had hazard stickers on the coolers inside, room temperature boxes, and on our tote bags.
Want a bio hazardous accident, think of one of the courier vehicles getting smashed, not just one, not just two or 6, but maybe a hundred or more different people's specimens potentially busted open, all of which could be carrying who-knows-what? After all, a lot of them were sent in because there was suspicion of something present, or known illness to be diagnosed.
Even if they weren't hazardous, icky stuff, blood, urine, stool specimens, not stuff you'd want to get into.
Those courier vehicles get stolen every so often, too, often with specimens in them. Not fun.