I see Jon Arryn as more of an Eisenhower role – he’s not the field commander, he’s there to focus on logistics, politics, and grand strategy. You can definitely see his hand in the Rebellion – he was the one who raised the banners and sent his charges from Gulltown to raise their levies, he was the one who kept the Tullys in the fold, and he was the one dictating peace terms in the end.
During the Greyjoy Rebellion, he likely remained in King’s Landing to coordinate a rather complicated war effort – coordination between the Crownlands, Stormlands, Riverlands, North, Westerlands, and Reach, logistics and supply for an amphibious landing by a royal army, etc. – so that Robert could finally have some fun in the field.