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I am confused about why the light released from the moment when gas first formed was so dominant in comparison to the light released afterwards. Why isn't the CMB in interference with a series of cooling blackbodies as the universe continued to expand and cool?

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It is. They're called galaxies and stars and we see them against the cosmic microwave background. However, they aren't (all) cooling because of the energy sources they contain.

But perhaps that isn't what you mean. When the universe become transparent it also means that the amount of radiation emitted by the H atoms afterwards is (a) much less and (b) won't have a blackbody spectrum. The two things go together. Blackbodies are the most effective radiators but they must be completely opaque, by definition.

There is also much less radiation emitted after recombination because there are about $10^9$ microwave background photons for every proton in the universe.

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