There are a few ways to think about this. Paid sick days are much better than widespread illness. One person being sick is better than many staff members being sick (and definitely better than customers/visitors/patrons being sick!)
Think about how many times a waiter or waitress touches utensils and/or plates - germs can spread easily that way. Also, chefs and food servers in restaurants can contaminate food if they're sick (by sneezing, etc.), increasing the chances for customers to contract the same germs. Another way to look at this question is: what if a chef is too tired or run down? They need to be home and rest, or they may make mistakes with cooking food that may cause serious illness in patrons.
Yes, people are getting sick due to the lack of paid sick days for food service employees. I believe unhealthy food service employees are risking the well being of their fellow team members and customers by going to work sick. Studies have shown that paid sick days reduces emergency care and virus outbreaks. Businesses may lose productivity due to paid sick days, but will gain by reducing their turnover rate.
Many companies are losing a large amount of money because they believe that they are losing productivity when a worker stays home sick. While they do lose some productivity, the worker is less capable of infecting co-workers. If many co-workers become sick, their ability to remain focused and healthy is compromised. Some companies are starting to turn this around, encouraging workers to stay home, even sending a catering service to their residence! While perhaps extreme, this preventative mentality will improve worker moral, decrease productivity loss, and even save lives.
Yes, because not getting paid for sick days decrease our productivity and make our immune system less ready to defeat bacteria and/or diseases. One should feel 100% in order to give his 100%. Studies have shown that there is a strong correlation between workers state of mind and appreciating their effort and giving them the amount of money the deserve.
Fast food workers, with their barely liveable wages and poor health coverage can not afford to take days off when sick. This in turn has a sick person preparing food for you to ingest. Any number of viruses can be transferred this way. So instead of one person being sick we have several others due to spread of germs.