The reason that this could affect the running of the clinic is that these doctors will most likely have a hard time approaching their job from a long-term perspective. If there are senior support staff who are engaged in long-range planning, this is less likely to be a problem. However, the head of the clinic is always going to be someone whose vision is going to be limited by their short term in office.
A major operation like the Cleveland Clinic needs to have a long-range vision. It needs to be able to plan for where it wants to go over the next few years. If its chief executives are always on one-year contracts, it will be difficult to achieve this. The chief executives will typically want to have short-term accomplishments that will make their tenure look like it was productive. This may become problematic if they pursue short-term goals and fail to plan for the long term.
It is important to have continuity and vision at the top of an organization. It may well be difficult to get this if the chief executives are always on a one-year contract.