Not necessarily. All he would be was a weapon to be wielded as his sister desired. Gareth Bryne did achieve fame in his position as Captain-General, but this was largely due to his own qualities, not his position. In fact, the only reason he had that position was because of those qualities. I don't see why Gawyn would harbour any particular expectation of greatness in that regard, since he would consider Galad to be his superior in the matter. Besides, if that were important to him, he would have returned to Andor for the succession, wouldn't he?