This can depend on a lot of things, like weather, cleanliness of the conditions, etc.
I've killed animals in our summer down here and put their meat in a plastic bag and hiked it out, only to get it in a breathable bag and on ice after a few hours. It's been fine. I've also killed animals in cold enough environments to hand the quarters with the skin on and let them cool down for a few days and then put them in meat bags like that and packed them out.
My preference will always be for putting meat in a breathable bag, and for this reason, I prefer packing out meat in bags that have a meat shelf, so you can keep it separate to the rest of your stuff. I have packed meat out in a bag without a meat shelf plenty of times, too, and I used heavy duty plastic bags for this purpose. The meat had already cooled down overnight in these instances so there wasn't a huge issue needing to let it breathe.
I understand the implications over there with not wanting meat to spoil and all that, but it's not as complicated as it's made out to be. If putting the meat in a plastic bag is something you decide to do, because you value cleanliness over breath ability, then go for it. Just make sure you get it cool as fast as possible once you get to where you're packing.
Having said all that, a bit of blood on your bag and on you is not the end of the world. Hell, even a bit of blood on most of your gear isn't the end of the world. When I make decisions about packing meat out, it is entirely to do with how cool the meat is, how cool the conditions are, and how dirty things are likely to get in terms of dirt, leaves, etc. Getting blood somewhere is never in my thoughts.