My mostly on my gut feeling based counter argument would be:

• So what? You are cooking the water afterwards cross-contamination between water samples isn’t a huge deal. Additionally, Filters (especially things like cloth) are cleanable and potentially sterilisable via cooking.

• cooking is a violent process, grinding down particles, lessening the effectiveness of the filter. So you are potentially worse off, for no real gain.

• You can’t always cook. Sometimes you have to sterilise water another way. E.g. via Exposure to as much UV/Sunlight as possible. Particles in the water lessen or prevent this from happening