Some might put them in the fridge when they close or even have many extra so that they can be rotated in and out. But, given the frequent state of ice machines in restaurants, it wouldn't surprise me if the way condiments are handled is less than ideal.

Though, on the other hand, if it's not resulting in a bunch of cases of food poisoning, maybe it serves as immune system training that just gives you an ick feeling when you realize what's been going on this whole time.

Many places use those individually sealed packets, too. Ketchup bottles usually say "refrigerate after opening", so those things are probably fine, other than maybe plastic content.