What is the current status of geophysical global cooling theory?
The effects one can see on other planets (and many of the icy moons in the solar system) are typically faults and scarps due to the fact that during cooling, these bodies contracted. Since the surface was solid, the only way to accommodate the contraction is through faults.

But Earth's surface is not solid on long time scales. We have plates that move, scrape, and subduct. If you shrink the perimeter of Earth slowly, you don't get compression faults. Rather, you're just going to accelerate subduction a little bit and maybe slow down mid-ocean ridges a bit. 

In the end, the point is that the system of plates that covers Earth can accommodate slow shrinkage of Earth without producing visible faults.