I don't think "all" corals “need†to be fed ... You can keep brains and not target feed them and they will be fine.
All corals need to eat--exactly how much depends on many factors. Whether they will get enough food from leftovers from feeding fish, from fish waste, etc. or not again depends on many factors. It would be wise to actively try to make coral-appropriate food available in the tank though.
Lack of feeding wouldn't be the reason for the recession, you've likely got something else going on with your tank or the brain's placement that's causing the recession. Where do you have it in the tank, what are the flow and light like, and what are your water parameters.
It doesn't hurt to feed corals, but if the coral is "happy" in your tank you shouldn't have to.
Lack of feeding definitely could be a reason for recession and I've seen this happen to many corals. However, there are a lot of other factors that can cause recession besides starvation and most of those are often more common. How "happy" a coral is in a tank though has nothing to do with whether or not it needs food. That would be a bit like saying that if a person walks their dog twice a day and pets them often (keeping them happy and stress-free) that they don't have to feed them. Yes, yes they really do.
Chris