Cats are not truly "independent". They enjoy time spent with people; they just don't express their affection in the same way that dogs do. They enjoy companionship and interaction. This is why feral cats live in colonies rather than in a solitary state of existence.
Food becomes stale and tasteless and water becomes flat tasting after standing out for an extended period of time. Although pets have no choice but to eat and drink what is left out for them, they really don't enjoy doing so anymore than we would enjoy eating and drinking under these conditions.
Cats are very fastidious animals, as evidenced by their constant self-grooming, and could resort to house-soiling if a litter box goes too long without being cleaned. Once this behavior starts, it can be difficult if not impossible to correct!
There is also the possibility of serious injury or illness going undetected if there is no one coming in to check on the welfare of your pet. A friend's cat was playing with the cord on a set of mini-blinds and somehow managed to get the cord wrapped tightly around his front leg. When she found him he was hanging by the leg and had broken several teeth trying to chew his way out of his predicament. There was genuine concern that he might lose the leg due to permanent nerve damage or tissue death due to lack of blood circulation. Fortunately, he recovered the full use of his leg, and his mouth healed well, although the broken teeth eventually had to be removed. Imagine the tragic outcome if he had not been found for several days.
Another possibility is that of a male cat "blocking", or being unable to urinate. Surprisingly, many cat owners have no knowledge that his can happen. Unless this situation is corrected within several hours of onset of the condition, the cat faces almost a certain and miserable death.
These are only a few of the reasons why a cat should never be left without someone checking on its welfare at least every twenty-four hours. You obviously love your pet, or you wouldn't have taken the time to read this post. Have someone come by at least once a day to supply fresh food and water, scoop the litter box, and spend some time with your pet. Even if your cat isn't sociable, its well-being is being checked upon. Peace of mind is priceless, and so is your pet's life!