Non/existence is all about memory, though. I don't remember what I was up to before I was born, but I don't remember the first few years afterwards either. I only have other people's word that I existed, and a few cute baby pics. Stretching the point, I don't remember what I was doing at 3AM today either, although sleeping is a strong evidentiary contender.
I would strongly disagree with that.
Existence (of a person) is a matter of a working (I wouldn't even say functioning) neural matter, or something equivalent. It doesn't matter whether any of the past workings are remembered.
Like you said, we remember nothing from a time well after we were born, even though we are certain that babies of those ages do exist, do have working brains - feel pain, feel satisfied, even seem to have a sense of self and of relationships.
Similarly with people with severe dementia: They cease to remember pretty much any new thing, by and by lose functions (that's why I would not require brain to be "functioning", as a brain and body can survive biologically without strictly the need for any mental functions related to personhood, consciousness etc), by and by also lose existing memories, lose speech even - and yet I insist they exist until the day they die.
Similarly with people in a coma whose consciousness and, presumably, memory functions are on hold and who die without having regained such upper brain functions: I would argue here that their brains have (probably) retained at least a subset of potential mental capabilities (includin memory), even if those have not been called upon for a while.
If on the other hand you keep a body alive that has lost its head (say, you are interrupted in the middle of a head transplantation, or really a body transplantation), then the brainless body probably does not "exist" as a human person (this is assuming that none of the neural fabrics elsewhere in the body - spine and intestines come to mind - can form a mental image of self).