An assembly is provided for an aircraft propulsion system. This assembly includes a sun gear, a ring gear, a plurality of intermediate gears, a carrier, a first brake and a second brake. The sun gear is rotatable about a centerline axis. The ring gear circumscribes the sun gear and is rotatable about the centerline axis. The intermediate gears are arranged circumferentially about the centerline axis. Each of the intermediate gears is meshed between the sun gear and the ring gear. The carrier is rotatable about the centerline axis. Each of the intermediate gears is rotatably mounted to the carrier. The first brake is configured to slow and/or stop rotation of the ring gear about the centerline axis. The second brake is configured to slow and/or stop rotation of the carrier about the centerline axis.