I think you are conflating thermal conductivity (how well thermal energy propagates through a material, measured in W/kg*K) and thermal capacity (how much energy a given amount of a material can hold, usually measured in J/kg*K).

Water has high values in both btw. So it takes a lot of energy to heat up and cool down, but it also exchanges energy quickly within itself and to it's surroundings.

Now instinctively I thought that water would have a higher thermal conductivity than ice, because Iglus insulate so well. But snow isn't ice (crazy I know:D) and it turns out ice has about a 4 times higher thermal conductivity than water at 0°C.

Therefore: If you cover the ice in a thin layer of water, this should slow down the melting. But if you put the ice into a decent sized container with water, where the total surface area of the mixture becomes more than 4x larger than the surface area of the ice it should speed up the melting process. This effect should also increase the further along the melting process you are, since the surface area of the ice will decrease (less of it left), where as the surface area of the mixture remains mostly equal (ice has a bit less density I know, but small effect).

So I learnt something: Solids generally conduct heat better than liquids. But the original point, where it depends on the container and it's conductive surface area still mostly remains valid.