In English there is a big difference between "They hotly said 'Hello'" [they were mad] and "They warmly said 'Hello'" [they were friendly]. So...maybe varme should NOT be accepted as "hot" or "hotly" in other situations? Am I wrong?
Except that, in English at least, and often in Esperanto, the adverb typically goes before the word it is modifying.
"Hello" is not the word being modified here, "said" is.
Nice try though, and thank you for including the exact sentence you were asking about.