Eggs, in a baking recipe, are usually there to make the matrix/web of the structure. The protein provides a sort of elastic glue which binds the structure together. Yolks can usually be replaced by another fat source, but the whites are difficult to replace. If you think of what happens when you whip eggwhites, they form a structure around little air pockets. There’s not a lot of other ingredients that can achieve this.
I wish I could remember the technical name for the ‘bean water’ that was being experimented with for making vegan meringues. I think it would be the best substitute if you are omitting egg white.
Mind you, eggwhite is almost pure protein. A lot of folk that have allergies to egg only have a problem with the yolk, not the white. Of course, that is something that would require medical testing.