What [if anything] is special about the boiling process of purifying water (excluding temperature)?
There nothing special about boiling that is doing anything.  For example, you can put a beaker of water under vacuum and get the water to boil at room temperature, but that won't kill any bacteria or render any viruses impotent.

It's purely a heat thing.  You want to destroy any pathogens that may cause illness.  Sufficient heat will denature proteins in a virus and kill bacteria, etc.  Boiling simply gives you a visual cue that you've reached a very high temperature (so you don't need a thermometer) and the ~1 minute is just to ensure that you sustain a high temperature long enough to destroy any bacteria or viruses.