Residual policy learning (RPL), in which a learned policy refines a static base policy using deep reinforcement learning (DRL), has shown strong performance across various robotic applications. Its effectiveness is particularly evident in autonomous racing, a domain that serves as a challenging benchmark for real-world DRL. However, deploying RPL-based controllers introduces system complexity and increases inference latency. We address this by introducing an extension of RPL named attenuated residual policy optimization ($α$-RPO). Unlike standard RPL, $α$-RPO yields a standalone neural policy by progressively attenuating the base policy, which initially serves to bootstrap learning. Furthermore, this mechanism enables a form of privileged learning, where the base policy is permitted to use sensor modalities not required for final deployment. We design $α$-RPO to integrate seamlessly with PPO, ensuring that the attenuated influence of the base controller is dynamically compensated during policy optimization. We evaluate $α$-RPO by building a framework for 1:10-scaled autonomous racing around it. In both simulation and zero-shot real-world transfer to Roboracer cars, $α$-RPO not only reduces system complexity but also improves driving performance compared to baselines - demonstrating its practicality for robotic deployment. Our code is available at: https://github.com/raphajaner/arpo_racing.