The item one wears doesn't have to be observed by others (think underwear)—it just has to be carried on the body. The idea is that you can go about your business, over time, with the item in place. In several other languages, there are verbs translatable as both carry and wear.
The thing with hand sanitizer is not that it stays on the hands as a protective layer but that it does its job of cleaning the skin and is then used up. That's why use is good but wear doesn't work.