You are over simplifying a little. Tanny wasn't just a figure head. He was the GM. He oversaw the scouting dept as well as ran the cap and led contract negotiations. Those are huge responsibilities.
I think you are referring to picking players, which is probably the most important thing. Tanny had a role but since his background was not based in personnel, he relied heavily on his staff of scouts and coaches to make personnel decisions. Probably more so than other GMs. Rex pretty much got every player he ever wanted in the draft and FA as long as we could afford them.