Different species of anemones have different feeding and light requirements. I would say that most anemones do certainly benefit from feeding, and they certainly eat (or in the case of hosting anemones, fed) in the wild. Carpet anemones are some of the most aggressive, and can and will eat sizable fish should the fish venture too close.
For most, I would say that the filter feeder foods are too fine. It just doesn't quite work with their anatomy, which consists of tiny harpoon like structures. It would seem to me that those particles are too fine to be captured, but if the carpet is happy, and doesn't want the larger food, then so be it. It might very well be satisfied with the lighting. However, don't be surprised if it eats something some day.
Condylactus anemones, which are from the Caribbean and are not typically a hosting anemone, are nonetheless some of the easiest to care for. IME, they are far more reliant on feeding than on lighting. These guys should be spot fed with MEATY things, silversides, krill, squid, etc. Quite often in fact- at least a couple of times a week.
They will "poop" by contracting and basically pumping all of the water out of their body, making them deflate. They will reinflate later. This happens in all anemones from time to time.
All other anemones in the trade, beginning with long tentacle and bubble (the easiest) to carpets and sebaes (some of the most difficult) need stronger lighting. They do most definitely benefit from feeding with meaty things, which again, they would get in the wild. Most people who report splitting of their anemones do provide them with food, unless they have clownfish, which may feed the anemone themselves. Indeed, this behavior helps to insure that the anemone stays healthy, and therefor remains a good and safe home for the fish.
However, never assume your clown is doing this, as they are notorious for shoving food into the anemone one minute, and running in to eat it themselves the next.
Simply, if the anemone is not hungry, and does not want the food, it will spit it back out. I think it is best to give them proper light, and proper feeding.
Also be sure to change out the light bulbs if they are more than a year old.