You are talking about two different types of そう which are grammatically and etymologically different. Since they are unrelated to each other to begin with, the meaning of one cannot explain the meaning of the other.
そう as an adverb meaning "so" derived from the archaic adverb 然 (さ), which meant "so" or "that way". On the other hand, the etymology of そう(だ) meaning "to sound/look" is not very clear, but dictionaries say it's probably from either 様 or 相.
When you see そうです preceded by nothing, this そう means "so". If そう is preceded by a masu-stem or an adjective, it means "looks/sounds". (And of course there is
そう for hearsay, too.)