It can’t be used figuratively and retain any meaning.

It literally can, because context doesn't come from words alone. This is the danger of only considering semantics without also considering pragmatics. Context can come from words, but it can also come from situations, body language, italics in writing, anywhere humans can discern meaning. Putting "literally" before a situation that you know from your world knowledge cannot possibly be true literally provides the context for interpreting "literally" as "figuratively".

Check out the Gricean Maxims for more about this, specifically the "flouting maxims" section. Basically, if a person says something that doesn't appear to be truthful under the most immediately apparent meaning, speakers automatically search for a context under which that interpretation becomes sensical. This is how hyperbole, irony, sarcasm, basically all uses of language that require interpretation beyond surface-level semantics, works.

In short, using "literally" this way is literally the same thing as telling your clumsy friend that their gracefulness is impressive after they slip and fall.

Also, "literally" has been used this way in English since at least the 1700s, by writers like Fitzgerald, Joyce, Bronte, and Dickens, to name a few, and it's in literally all major dictionaries (not that that should matter, but you seem the type that would be important to).

Most importantly, though, people use it productively in fluent speech, which is literally the only reasonable (and objective) measurement of what is and is not grammatical for a given speaker.

the word “literally” is supposed to mean “without hyperbole or sarcasm”

According to what objective standard? Good luck finding one outside of maaaybe special-use technical jargon. Here's the literal dictionary's take on it, if you'd prefer to go that direction with the conversation.

Words mean what their speakers fluently use them to mean - anything else is just linguistic snobbery masquerading as concern.