A good question but out of my depth. I asked Prof. Warren Kimball, author of the Churchill-Roosevelt Correspondence and other books on their wartime relationship, who replies as follows:

Much of what Dr. Capet writes is accurate. Cordell Hull was initially appointed for his strong political connections with Congress, having served served eleven terms in the House of Representatives (1907–21, 1923–31). Elected to the Senate in 1930, he resigned in 1933 to become Secretary of State. He was a solid New Dealer and a Wilsonian internationalist who won the Nobel Prize in 1945 for his role in creating the United Nations. But he was never part of FDR’s inner circle.  He did promote Roosevelt’s “Good Neighbor Policy” toward South America, and supported aid to Britain in the prewar years. But from the outset, President Roosevelt preferred to use Sumner Welles as his first-line diplomat. Hull did not attend any of the major wartime conferences, while Welles participated in the ones held in the U.S. and Canada before his resignation in 1943. Thereafter, Averell Harriman became FDR’s major representative in wartime Europe.

Hull’s one great moment came when he flew to Moscow in autumn 1943 to meet with his fellow foreign ministers, Eden and Molotov. Hull and Roosevelt had differences about how to structure the postwar world, particularly over FDR’s “four policemen” approach, which Hull feared could become little more than spheres of influence. But ego more than substance was what set them apart. Hull’s pride was well stroked by his appointment to head the American delegation. Economic liberalism, decolonization, suspicion of Britain, China as a balance-weight in Asia, and postwar cooperation were broad concepts he and the President could agree on—and by then the goal was to get the British and Soviets to agree to the principles, not the details. 

Moreover, Hull was disinterested in military affairs and would not get enmeshed in the Second Front or Mediterranean strategy disputes. Roosevelt met Hull on his return from the foreign ministers’ meeting, but barely listened to the Secretary’s report. As Hull commented somewhat forlornly in his memoirs, the President “was more interested in discussing the forthcoming conferences at Cairo and at Teheran,” than hearing anything more than highlights of the Moscow talks.