I think you’re correct that it smells gross. Medical tape is a good idea.
I’ve also used scotch tape and electrical tape in a pinch.
If I were trying to sell you duct tape I’d say something like “it’s so strong and it’s rigid enough that you can even use it as a makeshift splint.”
And I usually remove it after a short while just long enough for them to heal. But sometimes I forget and it sits there forever 