Storage-disaggregated databases have become the standard in the cloud due to many benefits, including improved resource utilization, reduced resource fragmentation, and the ability to independently and elastically scale compute and storage, ultimately leading to cost savings. This work focuses on OLTP databases. Examples include Amazon Aurora, Microsoft Socrates, and Neon. However, a significant limitation we have identified in storage-disaggregated databases is the long tail latency. This issue arises from the unique architecture of these databases, specifically the log-as-the-database design principle. Under this design, when a transaction is committed, only the logs are sent to the storage engine over the network to minimize data movement, while the actual pages are replayed on the storage side. Thus, certain page requests may encounter a lengthy log replay chain, which lead to long latency. In this paper, we introduce a novel technique called \textit{Replay-as-a-Service} (RaaS) to address the high tail latency issue in storage-disaggregated databases. The main idea behind RaaS is to decouple the log replay logic from the storage engine and make it as an independent service. This approach provides the flexibility to utilize idle servers or even dedicated servers within the cluster to efficiently execute the log replay. To enable this, we introduce a suite of techniques and optimizations to address key technical challenges. We have implemented the RaaS technique within OpenAurora, a popular open-source storage-disaggregated database based on PostgreSQL. Experiments on SysBench show that RaaS reduces P95 tail latency by 40.1% and improves the overall throughput by 75.9%.