The correct service that distributes client requests among various computing resources and ensures smooth failover in case of unhealthy resources is the Application Load Balancer. By monitoring the health of its registered targets, it can reroute traffic to operational targets in different locations, maintaining application availability and reliability. Route 53 is focused on DNS management and traffic routing policies but doesn't handle the actual traffic distribution across multiple compute resources. EC2 Auto Scaling is used to adjust the number of EC2 instances to meet changing demand, but it is not responsible for request distribution. Lambda executes code in response to events and manages the compute fleet required to run the code but is not involved in routing traffic or handling resource health checks for failover purposes.