The problem asks which property of a metal block primarily determines how readily heat is conducted through it when placed in a cooler liquid inside a calorimeter. To solve this, we need to understand the role of different properties of the metal.

Let's evaluate the given options:

1. The thermal conductivity of the metal block: Thermal conductivity is a measure of a material's ability to conduct heat. It dictates how quickly heat is transferred from the hot interior of the metal block to its cooler surface and subsequently to the surrounding liquid. Therefore, the rate of heat transfer through the metal block is primarily determined by its thermal conductivity.
2. The specific heat capacity of the metal block: Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. While it affects how much heat the block can store, it does not influence how fast the heat conducts through the block to its surface.
3. The internal density profile of the metal block: Density influences the mass per unit volume of the metal but does not directly affect thermal conduction. It is not a primary factor in determining heat conduction rate to the surface.
4. The absolute volume metrics of the metal block: The overall size or volume of the block may affect the total heat capacity but does not govern how quickly the heat moves through the material.

Based on this evaluation, the property that primarily determines how readily heat is conducted through the block to its surface is indeed the thermal conductivity. Thus, the correct answer is:

The thermal conductivity of the metal block

This conclusion is supported by the fundamental concept of heat transfer that emphasizes the role of thermal conductivity in facilitating heat flow in materials.