The biggest need that the ledger addresses is to provide an authoritative, globally consistent answer to this question: “What is the public key for identity X?” (Or what was it, at a particular point in time…) The answer to this question is the foundation of secure communication, of reliable digital signatures, and so forth. It is remarkably simple, and remarkably powerful. A related question is, “Where is the agent for identity X?”
You can’t use agents with confidence, without the ledger–because you have no way of knowing the authoritative endpoint of an agent. You can’t use digital signatures without a ledger, because you can’t test their appropriateness against a global source of truth.
The ledger is also required for the whole claims/proofs ecosystem, because without the ledger, there is no authoritative way for them to be signed, and no authoritative way to revoke.