They should but I've never known anyone have time off work for cold sores which is highly contagious. Where I work we all sign a form on starting for the company that states that we will not come into work for 48 hours following any diarrhea or vomitting too but I bet we are in the minority.
Indeed, especially if people are able to show a positive test for a particular illness. I don't want to get into the whole (covid isn't flu nonsense) but I think one thing we should take from the last 2 years is our attitudes towards Flu have been poor and have killed many vulnerable people. That also includes many vulnerable people who likely weren't conscious of how they could mitigate their own risks.