Edit: Mystery solved. Thermal pads seem to work fine thermally and insulate the chips electrically, at the same time.
The IC package is plated and electrically connected to local ground, which means that unless there's electrical insulation between the package and heatsink, you can cause shorts and break stuff. We use a 0.5mm 20W/k silicone pad to avoid shorts.
Doesn't silicone have low thermal conductivity? I would think electrically insulating and thermally conductive are tough material properties.