I see both sides of this. For the side of the non-hunter, I will relate what I have heard from my mom time and time again. She grew up outside of Chicago, as did I. So many time she told me about a couple of times she traveled with her family while young, and then again after marriage, seeing dead deer strapped to cars. The top, the trunk, even the hood. (I would think the heat from the engine would spoil the meat.). But she was horrified. Her first impression of hunting was a mix of my grandfather shooting pheasant as she watched, and seeing beautiful deer dead and strapped to cars in what she remembers as this endless procession of corpses. She even relayed this story to me again when I slipped up and mentioned I was going hunting.
Somehow, despite not being brought up with hunting, and hearing these stories, I hunt anyway. But I can see how this may not be a terrific impression for some people. The solution? I am not sure there really is one, because I believe in the rights of hunters, and when I see a dead deer on the back of a car in a parking lot, I am checking it out and congratulating the driver.
My own decision is that someday (a million years from now when I finally shoot one) is that when I have a deer to transport, I will stash it in the back of my suv. Maybe at the local store I will notice someone else's deer and have an opportunity to strike up a conversation and show mine off too. But if I had a car, not an SUV, on to the back it would have to go, in a tarp, out of respect for my mom and people like her.
On one hand I think people should be able to transport them however they deem fit to do so. On the other hand, don't want hunters to get any more of a bad rap. I am unsure where I stand as of now.
I wish it were so simple as to just do as you please. Because I would be the first one to want to pull into the local store where everyone congregates with a nice deer for all to see.
Oh, but to answer the initial question, no. I do not believe it should ever be illegal to be transporting a deer as you did! I think how you do it should only EVER be a matter of personal preference.
And then the moon, like to a silver bow new bent in Heaven