Please note that thermal conductivity is one of 2 big properties. The other is specific heat capacity i.e. how fast does it heat up.
You already noted that:
There are 2 kinds of specific heat capacity; the most common is gravimetric i.e. “how much heat does it take to up the temperature of 1 gram of substance by 1 K”. For us the most interesting is volumetric i.e. “how much heat does it take to up the temperature of 1 m³ of substance by 1 K” though both matter. Here heavier materials (like copper) have an edge.
Heat capacity, like conductivity, depends a lot on alloy. From my old notes:
Regular alu 6061 has thermal conductivity of 150 K/W and heat capacity of 2.42 J/mm³K. 6063 has 200 K/W and nearly the same 2.43 J/mm³K. Copper has 400 K/W with 3.44 J/mm³K. 7068 which I think is the overall best flashlight alu for being much tougher than 7075 with great thermals – has 190 K/W with 3 J/mm³K.