I'm sure they do. But in a conflict, especially in war, there needs to be a least one person that stands up and says "follow me".
While the Ionian Elders may decide as a group the best strategy and direction to take, there needs to be someone with the final say on the best tactic fulfill that strategy. To have multiple people at the very top of command would just lead to slow decisions, possible arguments, and miscommunication.
In politics, group thinking can work by itself with no central leader. In war, due to the need for quick decisions, a central leader is needed, at least on the battlefield.