I do think the idea behind cooling the rice down fast is to prevent the opportunity for undesirable bacteria (and maybe yeast) to settle and start growing. Interestingly, letting this happen is actually integral to some old methods of brewing (see below). Auld's method of water addition works pretty well from my experience, especially if you use more water (shift amount to refrigerate ahead of time from the total that is added at that stage). Any reason you don't want to do that? If you leave the rice out to cool, just making sure to use sterilized trays and cover with a clean cloth or plastic sheet should go some length to prevent unwanted microbes from settling in.
Your freezer/fridge methods might work for the smaller additions, but maybe less effective for the larger rice additions. I don't think 5 min in the fridge will be enough though, and at longer times in the fridge, rice may start to dehydrate.
Modern breweries in Japan use a machine that sucks up the cooked rice and blows it out into the fermentation tank, cooling it along the way. Or some put the cooked rice on a conveyor belt that moves the rice through jets of cool air. Just watched a video brewery tour from Terada Honke, one of the more "old school" shuzo in Japan and they just spread the rice out onto mats and let it cool in the ambient temp of the brewery. The toji explained that he doesn't like anything blowing on the rice, and prefers this gentler, slower process. They use ambient yeast and cultivate their own koji (highly unusual).
Brooklyn Kura uses a shop vac to pull air over the cooked rice to cool it down faster. On the Sake Discord, there's a sake making thread wherein someone is currently working on building a cooling table. Looks like a flat metal grate with a shop vac underneath to pull air.