0.84W/(mK) is OK for thermal grease. These greases are never too great at conducting heat (compare to aluminum at around 200W/(mK)), the key is they are much better than air gaps, so first make sure the surfaces are flat, smooth and free of dirt particles, then apply modest amount of thermal grease, and enough pressure so that excess grease squeezes out.
Better thermal pastes exist but definitely not used in general purpose electronics. They only make a small difference in special cases, some computer overclockers might benchmark some minor advantages.
By paper like thermally conductive material, you need to make a distinction between materials which are just thermally conductive, but still require the interface material such as the grease, as they are "hard" and don't fill the tiny gaps themselves; classic example would be a thin mica sheet. And then there are so called "thermal pads"/"silpads"/"gap pads", which are sheets of soft, sticky material. With those, you don't need the paste. They (not all!) offer proper electrical insulation at the same time, something which grease alone does not. As a compromise, thermal conductivity is not that great because the layers are thick (0.5mm would be fairly typical - way thicker than any grease layer, ever!).
For CRT repair, the most important aspect is to understand whether you need electrical isolation or not. If the original solution really was just grease between the component and heatsink (i.e., there is no mica sheet), then there is no isolation, and applying new grease is the correct solution.
If the original solution uses insulation layer, you need to work carefully, understanding what goes into maintaining that insulation.