The reason restaurants do not refrigerate ketchup is because the ketchup does not require any refrigeration.
Ketchup can sit out on the counter, table etc at room temperature without going bad.
I've never refrigerated ketchup, mustard or even mayo and never gotten sick from it.
Some foods require refrigeration to keep the food safe from growing bacteria but ketchup is safe outside of the refrigerator and will last 6 months or more before you use it.
You can refrigerate ketchup if you want to refrigerate the ketchup but it's not required.
Ketchup sits out at many restaurants and homes and it never goes bad.
The reason ketchup does not go bad at room temperature is the ketchup has a natural acidity which prevents bacteria from growing in the ketchup so the ketchup does not need refrigeration.
Refrigerating ketchup after opening helps keep it fresher longer but there's no requirement to refrigerate ketchup and the health department and health inspectors do not require the restaurants to refrigerate the ketchup.
It's mostly used up pretty quick in the restaurants anyway.