Do you think it would also benefit me if instead of stepping around the tree, hugging it, and trying to get one gaff in before stepping off with my other foot, would it be better if I just stepped up off the platform and once I'm on the tree make my way around to the side and start climbing down?
Yep, that's what I'd do. I'm speaking from an arborist standpoint. I did tree work for several years. I'm very comfortable with gaffs. Personally, I'd never "hug" the tree. Leaning away from the tree gives your gaffs a better angle to dig in. Hugging the tree makes a gaff out more likely. I'd step off the platform onto the tree just like stepping off the ground onto the tree, in an upward motion like I was climbing higher. Then I'd work my way around the tree, left or right either way. Then I'd start down. Once my lanyard got down to the platform, I'd remove the platform from the tree, drop it, clip it on your harness, or your lowering line & lower it, or whatever you do to get it off the tree, & then climb down.