When reading through The meaning of そう, I was thinking about そう's status as a 副詞. Certainly, it acts like many other 副詞 in that it can directly modify verbs.
However, I was curious about the usage そうだ -- a form that I think not all 副詞 support.
副詞 that directly work with だ:
こうだ、どうだ、ああだ (?) all work, along with some other things classified as 副詞, like なぜだ.
副詞 that work with だ by omission:
Then, I think there are other 副詞 that work, like きっとだ, たまにだ. But, I think these only work if you interpret something as being omitted, like きっと（できるの）だ or たまに（するの）だ, for example.
In other words, semantically, the 副詞 does not seem to be modifying the meaning of the copula.
副詞 that don't work with だ at all:
×決してだ, ×どうかだ, and probably more. These seem to share the trait that they need specific endings (決してするな, どうかしてください) which the だ gets in the way of.
Assuming my judgments are correct, this seems like somewhere where the 学校文法 framework is failing to explain what's going on fully. Is there a better way to think about these things? Are こう、そう、どう、なぜ actually also nouns? What is the actual syntax/theory of the cases where 副詞 work by omission?