Gas turbine engines lose power as altitude increases. However, drag also drops with increasing altitude, but the power (thrust) drops faster than drag. Usually, power available exceeds drag, enabling the aircraft to climb. Once the aircraft has climbed to an altitude that power (thrust) available drops to the point that it equals drag, the aircraft can no longer climb. That is the "ceiling" of the aircraft. As fuel is burned gross weight drops and with it drag. Eventually the power (thrust) available exceeds the drag sufficiently enabling the aircraft to climb again. That is why large aircraft "step climb" as they burn fuel.
There is an error in the following statement: "there would be a decrease in performance and a loss in power as there is more fuel than air in the mixture" The gas turbine's fuel control meters fuel flow to maintain a correct fuel/air mixture at all altitudes. But any engine's power output is dependent on mass airflow. As altitude goes up, air density goes down, and with it mass airflow. A supercharger system on a piston engine can compensate, up to a point.