What difference does it make in the organism's physiology/metabolism whether oxygen binds reversibly or not?
I think I see where you are coming from.  There is a misunderstanding here. 

Ice-fish have relatively lost their hemoglobin gene over time because they have adapted such a slow metabolism in the cold water where the oxygen is richer (oxygen content in cold water is higher).  The don't need any binding of oxygen to a carrier protein like hemoglobin because enough oxygen diffuses in from the water for the needs of their tissue. 

There is no binding - reversible or irreversible in their blood - that would imply that the oxygen is binding to a protein like hemoglobin or hemocyanin, which the fish does not have.   The oxygen is just diffusing into the fish from the water around it.  The circulatory system is helping by flushing the system out and keeping everything fresh as well.