When I lived in MO we had lots of deer road kills on the highways. The state police / highway patrol would pull the carcass onto the shoulder if there was still enough in one piece to drag by a leg. A DOT crew with a dump truck would come by now and again and shovel the remains into the truck. I sometimes suspected they liked to wait a couple weeks hoping the vultures & such would do the job for them.
I learned a new term today - I never knew those exploded tire bits were called anything. “Gators”. I like it. 
I think a large percentage of it isn’t ‘moved’ to the side of the road but rather stays in traffic until someone hits it over to the side of the road.
Roadies.
I hit a deer on an Arkansas two-lane. Called Game and Fish and they told me I could take it home. They were surprised when I declined.
Most of the small stuff just gets knocked to the side of the road over time by traffic. As a cyclist, I know this all too well…
Sometimes good Samaritans stop and help people move stuff that blows out of trucks and then they themselves get hit and killed:
Waste of meat. Even as a vegetarian, I would have took it home for friends/family.
Sometimes yes, sometimes no. There can be collateral damage when a tire on a rig blows. If a tire on a rear axle on a trailer goes, it’s probably going to damage the mud flap or the bracket that supports it, but still, a driver might not notice it until the next time he stops and does a walkaround. From my personal experience, an inside drive tire blew and put a hole in an air bag (air suspension). So I was getting low pressure readings on my air gauges on the dash. I didn’t know the airbag was holed when I pulled over to look at it, just that the tire was blown and I could hear air leaking. Luckily, I was just a few miles away from the truckstop I was planning to stop at for the evening anyway and I did have enough pressure to get there safely.
I called it in and they sent a service truck to replace the tire. He discovered the hole in the airbag, and of course, he didn’t have an airbag for a Volvo, so he managed to bypass the airline going to the airbag so my air pressure was good until I got to a place where I could get the airbag replaced and the airline unbypassed.