First of all, main thing for me is that having no "rotational support" on the hanging carcass annoys me and like said I'd avoid it if possible. "Because I want" is reason enough. I have also the priviledge to accompany some very experienced and efficient carcase handlers and watching them do it I know I'm not alone (they would also benefit from "rotational support" and some use workarounds to get it).
You must have some heavy winches if "cable set" keeps carcass somewhat steady... For the sake of discussion and learning, I try to make up an example. Please bear with me, not being a native speaker.
Trying to think what I do while skinning (the "spinning" annoyance applies also to butchering in my case) I think one of the things is that I'd like to use my off hand to pull the skin in horizontal direction, but that's impossible since it would spin the carcass. So I need to pull it in non-optimal (although still horizontal) direction. An example with deer would be, if the skinning at the chest area is not begun while in cradle (or at floor). So I'd like to pull the skin away from carcass as nearly at 180 degrees as possible (like I'd do in a cradle). But I'm restricted to pulling at 90 degrees or so. Do I get the job done, yes. Does this annoy me, yes also.
About the downward pressure, I don't skin deer from haunches to shoulders but other way around (if possible). When you get an opening at front legs, just push your "flat fist" through to spine area, and then same from the other side. Use your hand to separate the skin in upward motion towards the haunch. Much cleaner carcass and skin this way. We usually skin the carcass before dropping the stomach so quite warm etc. This might not work so good if you hang the carcass in skin and only do the skinning before butchering. And it doesn't work too good on old (whitetail) bucks etc. anyway.
And if you think an old buck is difficult, please try a wild boar. Getting the skin off is fighting for every inch. Ideally you'd have expertise and patience to peel the skin off and leave fat on carcass. But in most cases it's kind of hacksaw operation.