Ice and snow are both less dense than liquid water. Depending what type of snow and whether or not it's had a chance to thaw and refreeze, snow might even weigh less than 5lbs/ft^3 -- an order of magnitude less than your napkin math. I'd venture a guestimate that most snow people are shoveling sits somewhere around 15-20lb/ft^3, which is for snow that's settled for about a day, with wind, in below freezing temps.[0]

That having been said, you are not wrong. When we're talking about tarp-sized areas, any significant snowfall (i.e. anything not more quickly/easily cleared with a quick broom sweep) will get really heavy really quickly.

[0] There's a chart near the bottom of this page that describes types of snow/conditions, and lists imperial and metric measurements: https://roofonline.com/weights-measures/weight-of-snow/