From my reading of your message, the question is not whether the President and ED should be distinct entities, but whether the agency can function successfully without an ED.
Our organization recently went through a transition with a "Leadership Team" or Management Team while we searched for new ED. Basically the various managers within the agency worked collaboratively to assume the responsibilities of the ED. We appointed various individuals to take a lead in certain areas, including Board liaison, and worked through concensus. The Team was originally concieved to exist only until a new ED was found, but the Board considered keeping the team as a more permanent fixture. Ultimately, we did hire a new ED, but the experience was very positive. It gave the managers greater exposure to the functioning of the organization, empowered them to take responsibility and make meaningful decisions, and cross-trained staff in key areas. It also had its negative points though - it was incredibly time-consuming and egos became more of an issue than should have been the case.
Team management is not an unheard of prospect. Business cases demonstrate the possibility o fsuccess, but the personalities, structure of the org, and skills of staff have to be just right. With only 6 staff members, this may not be the case.