"... when they start the leaving process." We do get plenty of warning that it'll happen so there is time to mentally prepare. A child's independence increases inch by inch from the moment it can move away from its parent(s). It's the parents' job to facilitate this as smoothly as possible, which includes preparing their own feelings and attitudes for the ultimate nest-leaving.
That's not to say the nestlings won't be missed sometimes, but it's not difficult to adapt, bit by bit, to the changes that take place throughout a child's life towards their leaving the nest.
Well, maybe it is difficult for some but, as people have pointed out, there are lots of ways to make it less difficult for oneself. So do it. Take a step. Join something. Start a hobby or study something. Get involved in something elseor with other people. That's what the kids do and that's what parents have to do too.
We've had eighteen years, more or less, to get used to the idea.