There are likely a 1,000 way to do sick time and personal time for companies. Every human will have a sick day, and at some point, have something to take care of in their life that is not work related.
I have only worked for small family owned companies. I just went and talked to the owner and never have dealt with an HR department, ever. Now I have a mean slave driver boss that makes me fill out all sorts of paperwork and pays me only in beer and lets me drive his cars.
Personal time or trips to the dentist are always easy arrange since they are his teeth.