I think the existence of an identity isn't very important. You can adjoin an identity to any semigroup to turn it into a monoid.
In any case, doesn't adding more properties make a theory simpler? For example the classification of finite simple groups is much harder than the classification of finite simple commutative groups.
In any case, doesn't adding more properties make a theory simpler? For example the classification of finite simple groups is much harder than the classification of finite simple commutative groups.