After Irving Copi discusses arguments generally he discusses arguments using categorical propositions. These are (page 177)
that special kind of argument called deduction. A deductive argument is one whose premisses are claimed to provide conclusive grounds for the truth of its conclusion. Every deductive argument is either valid or invalid: valid if it is impossible for its premisses to be true without its conclusion being true also, invalid otherwise.
To get such an argument Aristotle used
only propositions of a special kind, called categorical propositions....Propositions of this kind can be analyzed as being about classes, affirming or denying that a class S is included in a class P, either in whole or in part.
So, it is not that "everything is a subject and a predicate". Only a "special kind" of proposition has this pattern.
As for "why do we need it at all", if we can write an argument using such categorical propositions and can show that deductive argument is valid, then we may be able to obtain "conclusive grounds for the truth of its conclusion".
Copi, I. M. Introduction to Logic. Sixth Edition. (1982) Macmillan.