As it happens, that is the technique I use.

I am not boiling the water to sterilise it but to delay the cooling. I would boil water from ice to sterilise it (If possible) because it's remarkable how many pathogens can survive freezing. At least short term. And chemical sterilisation is problematic at those temperatures.

I should have mentioned that it pays to stitch a small (4-6") length of cord to the bottom of the net. It makes directing the water into the cup easier.

The moment it hits freezing you are into Arctic/Mountain Survival techniques. It's not an environment that you can travel "light" in.

If it's a high mountain or ice field then your down to the between layers method.

If worse comes to worse, I would drink the water and take my chances.
Hypothermia as a result of dehydration is a far more immediate threat than any water born pathogene.