Hot tub cover for snow

However, my hands seem to feel more comfortable being exposed to winter air than to snow

Air (still air) is a better insulator than snow. In fact the insulating properties of snow are due to the fact is has a lot of trapped air within it. The other component of snow, water, is not noted for being a good insulator.

The trouble with using air as an insulator is that it's hard to keep the air still. Even if there is no wind the convection currents caused by your hands/whatever/tub will cause the air to move. And as the air flows away it carries away heat and brings fresh cold air into contact with the object. Though your hands may feel warmer in still air then in snow, I bet they'd feel warmer in snow than in a 100mph blizzard.

If you look at how animals use snow burrows they typically rely on their fur (or blubber if they're seals) as the primary insulator. The temperature immediately outside them will typically be minus a few degrees C. Where the snow comes in is by insulating them from the -40°C gale that's howling outside their burrow.

In the case of your hot tub, assuming the polystyrene lid you mention can provide enough insulation that the temperature above it stays below 0°C then a layer of snow would add more insulation if the external temperature is well below zero. If the external temperature is only around zero then the layer of snow will achieve little.