What is the primary advantage of using an orbiter like Mars Odyssey as a data relay instead of direct HGA communication to Earth?

It conserves the rover’s battery life by allowing the use of a low-power UHF transmitter.

The core reason for employing orbiters as intermediaries is power conservation for the rover. Relying solely on the High Gain Antenna (HGA) for direct communication to Earth would necessitate significantly more power than the mission budget permits, severely impacting the rover's ability to perform driving and scientific operations. By transmitting data in short bursts using the low-power UHF transmitter to an overhead orbiter, the rover manages its power budget effectively. The orbiter, equipped with its own more powerful transmitters, then handles the long-distance, high-power relay of the stored data back to Earth using the X-band.