Ah yes, a nice theoretical discussion on energy transfer.
The point about change of state is very true, this does occur with no temp change. Change of state from ice to water absorbs 75 times the energy used to raise the water temp 1 deg.(based on mass)
My point is if you get a block of long term ice from a deep freeze, it is at approx -20degC. A change of state will NOT occur in pure fresh water until 0deg in perfect circumstances. Surely if the ice bulk is -20 and the water is 0 then removing the ice from the water will slow the thaw.
You point about thermal contact is also true. That is why party ice does not last because the huge surface area versus relatively small mass enables quick heat transfer and therefore melting of the ice. Ideally if you could freeze a sphere of water in space, due to the minimal surface area vs greatest mass, it would stay frozen for longer than equivalent mass of ice in any other shape.
In refrigeration, ice is actually an insulator to heat transfer from cooling coils and needs to be removed by automatic means. When the automation fails or cant keep up, water is used to thaw the ice because of the heat transfer through contact.
Back to the esky situation, if possible, a large sphere of ice suspended above the melt water is the perfect solution
I still stick by the removal of melt water from the esky if you have still frozen foodstuffs in there.