You're right - Cu's thermal conductivity is higher. I had it exactly backwards.

The testing data I've seen show that even toothpaste will help conduct the heat between a CPU and a heatsink - no metal required. The real issue is filling the irregularities in the surface of the two metal surfaces.

The thermal characteristics of the choice of metal in the goop do make a difference, but it shows up as perhaps a degree or two C in CPU temp. Airflow and keeping the heat sink clean make bigger differences, on the whole. And, when we examine the choice of metals in many heatsinks, the metal is usually aluminum. If we wanted to wick heat away from a chip as quickly as possible, we'd use silver.