Maybe there's different kinds of Louisiana swampland, but the two bits of the Lafitte Preserve I've been in were pretty clean and clear. I'd drink from 'em, no worries. Sure bad stuff can happen, but bad stuff has already happened and people drank water for a long, long time before filters.
A couple of possible problems with that. We've become civilized enough that most of us are simply not able to handle the critters in most natural water because we're always drinking treated water, so our bodies aren't adapted. Even when they are, if you take a trip to a new locale with different critters in the water, you have to start adaptation all over again -- e.g., don't drink the water in Tijuana. I remember taking a hydrology class in college that included urban water and a field trip to water treatment facilities. Apparently our wastewater that goes through tertiary treatment is cleaner (or perhaps more accurately, more sterile) than most Mexican tap water.
I'm sure if you take clear water from a fast running source, you might be able to beat the odds, and if I had no other choice, I might do it. On the other hand carrying even a survival straw or clear plastic bag, or just carrying your water in a clear plastic bottle will let you do some filtering and decrease your chances of Montezuma's (or in this case Napoleon's) revenge. But your example of using fire is probably still the safest way to be sure.
Tangent: I recently caught an episode of a show called "Dual Survival". Kind of contrived, but it has an interesting premise of teaming a survivalist with a military background and training with a hippie guy that's into "aboriginal living" survivalism. The hippie guy walks everywhere barefoot and is into living with minimal modern inputs. In the episode I caught, they were "stranded" in Arizona, and one of their dilemmas was water. The hippie wanted to drink out of some scummy standing water because early humans would have, and all the larvae and insects and stuff would have been good protein. The military guy practically retched at the thought and would have none of it. Apparently the hippie drank crappy water all the time so his body was resistant to it.
