Side note: the study of monoids isn't a very big field in mathematics, because they're too simple to say too much about them. But if you modify them just a bit, either in the direction of groups or in the direction of categories, you get massive fields of study.
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.
