I have never treated my melted snow, nor do I know anyone who does. In fact, when I am on a backcountry ski trip I also do not treat any running water I may find. Noproblems yet! I doubt that ther is liekly to be any bacterial or viral contamination of snow on the ground, with the exception of summer snow with that reddish tinge in ti, which is an algae, I believe. New-fallen snow is essentially distilled water - it had to eavporate to get into the clouds, after all. I suppose it could pick up some chemical pollutants while in vapor form, but if it does, then I'm breathing that same stuff, I suppose.
And yes, I do add snow to my bottle during the day, and I also just grab a handful of snow as I plod along, Or if I am lucky I get to lick the fresh stuff off the needles of a pine tree. That's the best.