Many of the claims in the video are rubbish. Ozonation for disinfection is nothing new- it results in less flavour but has no residual effect, i.e. it is fine for initial "shock" disinfection but useless for protecting water from recontamination once it has left the treatment plant. The anti-cancer claims made in that video are a distortion in my view- they made no claim nor did they provide any evidence that cancer cells are killed preferrentially to non-cancerous cells, which makes sense given that ozone is a GENERAL toxicant. Drinking freshly ozonated water (right out of the ozonation tank) which an idiot in the video was encouraged to so, is dangerous- ozone is toxic and damaging to tissues whether it is ingested in water or in air. But since ozone has a short half-life in water (generally on the order of minutes), ozonated water would soon contain no detectable ozone and hence would be safe to drink.
I didn't hear a claim in this video about making radicals from bubble collapse etc. Those claims are frequently made about cavitation generated by ultrasound or other means. I've seen lots of these claims over the years but very little evidence for the efficacy of these devices as efficient generators of free radicals for water treatment, relative to other catalytic or radiation-stimulated means (i.e. UV peroxide etc.).
Making very small bubbles doesn't matter much except if you have an extremely fast reaction taking up the dissolved species, especially when self-reaction can destroy your feed gas if it hangs around too long. That can be the case with ozonation, but isn't usually the case even with ozone and is rarely the case when air is used as a source of reagent oxygen. There are exceptions of course. In aerobic water treatment, there are several things going on at the same time- aeration provides both oxygen and liquid/solid mixing in those systems, and fouling is also potentially a big issue. When comparing the efficacy of a particular aerator, it is important to control for these factors so you can really understand what's going on.
If the bubbles have to hang around for a long time to give sufficient conversion of the feed gas before the gas leaves the tank, i.e. aeration in a deep tank, it doesn't matter much how fine you make the bubbles in the first place. Surface tension and bulk mixing will rapidly determine the average size of the bubbles. Small bubbles will coalesce into larger ones quickly if there is insufficient liquid shear to tear them back into small bubbles again. Gas-liquid contact agitators are designed to do that.