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.