I was trying to mirror how someone would use the word “literally” which is, like my example with “waltzed” to say “I literally waltzed down the street.” I think, “I quickly walked down the street” would mean “I did not hesitate before walking down the street” in more cases than it would mean anything else.
Literally in that sentence, as I said, applies to “waltzed” but probably not to “down” (unless the word “down” is emphasized in speech) and almost certainly not to the street. If I change that to “I literally walked down the street” then we’d all be left guessing that the thing to be taken literally is the word “down” since there would be no reason we could think of to modify “walked” with “literally” when talking about going for a walk down the street (and again, in what sense other than a literal one could we mean “street”?). If “quickly” is applied to “walked won the street” then we already know what a quick walk down the street is like, so we can deal with that. If “literally” is applied to “walked down the street” we are left asking (and immediately subconsciously answering), “What part is literal?”. We need a lot of context to make sense of what people are saying.
But my main point is that in none of these cases would we guess that “literally” applies to “I”, and through that point to say that it doesn’t modify the entire sentence, but rather modifies the words in the sentence, and thus rebutting the “this is written in blue ink” analogy.
Having thought so more, though, I could accept the analogy and make another point: people saying that other people using literally for emphasis “don’t understand what ‘literally’ means” would be akin to someone looking at the sentence “this is written in blue ink” written in black ink and saying, “whoever wrote this doesn’t understand what ‘blue’ means.” That is, it seems an awfully unlikely, unimaginative and uncharitable interpretation of events.