Kurzgesagt has a video on it - The Deadliest Virus on Earth https://youtu.be/4u5I8GYB79Y?si=kpoOGGK8YUA-2P5k&t=378 (time coded to 6:18 where it talks about the lack of damage to the brain)

Part of it is that after it does its thing in the brain, it then travels to the salivary glands where it can then spread through a bite.