Did the phonetic component 䍃 spawn, or have anything to do with, the semantic narrowing from 言 to 謠?
"䍃" is a common 声旁, 谣 is a pictophonetic character (形声字), its pronunciation is from 䍃, 摇, 遥, 鹞 are pictophonetic characters from 䍃 too.
If not, what beget 謠's songful (#1, 2 below) and defamatory (#3) meanings that are more specific than 言?
What's 歌谣? Some words with tunes. What's 谣言? Fabricated words without facts. What's 言? Words. 言 is the meaningful part of pictophonetic character 谣.
For example, consider 謠言. Couldn't 謠言 have been originally compounded as 言謠?
"谣" is the "adjective" of "言". What type of words"言"? Uncertain or suspicious words. If it's 言谣, "言" and "谣" should be parallel.
Actually, I can't understand your exactly question because in fact, your question can be clearer.