I used to skin barehanded when I ran traplines and think I dodged the bullet, but then we also ate the beaver, muskrats and small coon we caught. When I started calling coyote I think it was the smell that turned me to use gloves. Fox never smelt like coyotes and being as delicate as they are it was hard to think of them as any kind of carriers. I'm older, smarter and I think a little more about not being indestructible than I did in my youth.
I carry nitrile gloves in my skinning box in the truck and keep them handy in the fur shed. I hate how slippery they are but wear cheap cotton gloves over them or synthetic gloves liners. I use the heavier nitrile gloves made for mechanics.
I keep a old first-aid box in the truck with a couple knives, a diamond sharpener, tail stripper, rope to hang them with, screw driver, nitrile and cotton gloves. I like to skin in the field as it is much easier to get rid of carcasses now that I live in town. I have one box in the truck and another on the MC other wise if I had one box it would always be in the vehicle that I'm not in.
I've never had a problem with transfers of fleas and ticks, coyotes with fleas get dumped in a bag after getting sprayed or powdered. Ticks I get covered with just walking through the brush. I don't go back to WI for early grouse, woodcock and trout fishing any longer.