The issue isn't Skin_Wrap modifier, it's how the newly created Skin modifier interprets the "zero pose" to count deformations from. Make sure you set the "ref frame" under "advanced parameters" in Skin modifier to a frame that has your bones in zero pose (matching your model), then toggle the "always deform" checkbox to…
Ok, Im not an animator.... Any hints or tips about why my mesh freaks out when I convert to skin? It seems to work fine with the bone based rig before conversion.