Aviate means to get the aircraft under control, straight and level as you say. Maintaining or correcting the position and direction of the aircraft, navigation, is of lower priority than keeping it under control. The idea behind the saying is to encourage the pilot to not prioritize maintaining altitude, heading, flight path, rules and regulations (or even following air traffic control instructions) over focusing on keeping the plane in the air. Communicating your situation, intent, and deviations to ATC and surrounding traffic is again a lower priority.
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