Why does a chemical rocket need so much propellant to reach orbit?

Answer: It must accelerate its remaining propellant as well as the payload

Once above the atmosphere, it needs fuel only to power its instruments — Leaving the atmosphere does not by itself put a spacecraft into orbit. It must also gain enough sideways speed, and rocket engines can provide that thrust in a vacuum. Propellant is not merely fuel for the instruments.

It must accelerate its remaining propellant as well as the payload ✓ — A rocket must accelerate both its payload and all the propellant still aboard. Reaching orbital speed therefore requires a large initial propellant mass relative to the final vehicle. Chemical rockets also carry an oxidizer rather than relying on atmospheric oxygen.

Most of its propellant leaves the engine without burning — Large propellant tanks are not mainly a sign that most propellant escapes unburned. Even an ideal rocket that uses its propellant efficiently needs a large mass ratio to gain orbital speed. Carrying the propellant adds to the mass that must be accelerated.