Septic "tanks" are actually a number of tanks in series. If they overflow, it's usually the early tank(s) that do it first and they're just as bad as a cesspool.
True cesspools rarely "overflow frequently" as they shouldn't be taking groundwater and they can't otherwise fill fast enough to do it "frequently". Running rainwater into a cesspool is a good way to do it though.
You don't just need to "have bacteria in" a septic tank, you need to have a population that's largely of the right sort, and you need to have them working for you. If you affect the oxygen availability in there then you might not kill them off, but you'll stop them working in favour of the anerobes (the hydrogen sulphide makers).
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