According to Chef In You, adding only one egg to a three-egg cake recipe means the resulting cake won't mix properly or will be too dry and will not rise properly. Eggs serve essential roles in some baking recipes; therefore, use egg substitutes if there are not enough fresh eggs.
Chef In You states that eggs are used in recipes to add moisture and richness, to provide leavening and to act as a binder. Eggs may be included in a given recipe for any or all of these reasons, or just added for flavor. Cake recipes use eggs for all of these reasons.