It is true that pathogenic organisms will be killed by bringing water to boil.
However, this is not gauranteed to destroy toxins produced by organisms. For example, Clostridium botulinum needs to be boiled for ten minutes to destroy the toxin. Cyanobacterial toxins aren't killed by boiling.
Like another poster said, bringing water to the boil will eliminate most safety risks.
I would suspect that this is where the practice of preferring moving water over still water comes in... If the water is moving and home to animal/plant life then I would have to assume that any microbial toxins are only present at low levels (along with any other introduced toxins).
However, this is not gauranteed to destroy toxins produced by organisms. For example, Clostridium botulinum needs to be boiled for ten minutes to destroy the toxin. Cyanobacterial toxins aren't killed by boiling.
Like another poster said, bringing water to the boil will eliminate most safety risks.