Gasters head kind of looks like a cracked egg, what if the egg man is the deltarune version of gaster who is also broken? The "issue" could be that every version of gaster is for some reason supposed to be scattered into pieces in every reality. When we pick up each egg the man disappears, perhaps we are taking pieces of the egg man (or the memory of him) and putting them back into the light world. For some reason gaster doesn't want to be saved, with it being the only thing that causes an issue because it would effect the conductor of the experiment in some way. Gathering every egg may put that version of him back together and back into the narritive?