Grapefruitseed extract toothpaste (Nutribiotic, citrucidal) is what I use. It does all you mention. I may try the bleach thing.
No, it does not. It is no where near sufficient. Trust the scientist who are LOOKING at the bacteria through microscopes after trying various antimicrobial approaches. No toothpaste is. You have to pack the backing soda down into the gums. The sodium bicarbonate kills the bacteria by a chemical process called crenation, the same process that kills slugs when you dump salt on them. Essentially, the conditions around the bacteria are rapidly changed to a hypertonic solution and the water inside the bacteria's cellular structure rapidly leaves its body by osmosis and it dies by desiccation almost instantly. In the case of the spirochetes, the do not have time to "ball up" and form their protective "spore" form to survive the harsh conditions.