You are right. The ozone only remains in the water for a short period of time. In fact, ozone is so reactive that it will even break itself down if left in a short period of time in a sterile environment. The amount of peroxide produced is low. On the other hand the amount of peroxide generated by the body to kill pathogens and cancer cells is also quite small. For example, natural killer (NK) cells actually kill cancer cells by attaching to the malignant cells and injecting them with peroxide. Because the malignant cells lack the antioxidant enzymes catalase, glutathione peroxidase, selenium methionine peroxidase, and superoxide dismutase (SOD) the malignant cells swell and rupture from the peroxide destroying them. Peroxides formed from SOD are also used to activate white blood cells. And peroxides formed from beneficial bacteria in the intestines and other parts of the body are used to control pathogens. All in small amounts.
I can speak from personal witness that 35% peroxide can do some serious damage to malignant tumors with very tiny amounts.