MR2,
Speculation: I think the number of situations where there is literally no law enforcement to respond are probably limited. In most situations you would prefer to avoid collecting evidence so as to not screw up by either contaminating it or the chain of custody.
Fact: Plastic bags retain moisture and heat which incubate bacteria which will then break down cells [and DNA] as food. DNA can also be degraded or by sunlight (think sunlight causes cancer via DNA mutations).
https://www.ncjrs.gov/pdffiles1/jr000249c.pdf is a pretty good summary.
I suggested paper then plastic because your primary mission is patient health, and I don't think the back of an ambulance on a rainy night is helpful for keeping fluids and other contaminants off of the weapon when you are distracted.