This seems to rest on a model of people as shallow, scripted puppets.
“Do you want my advice, or my sympathy?” is really asking: “which word-strings are your password today?” or “which kind of standard social script do you want to play out today?” or “can you help me navigate your NPC conversation tree today?”.
Personally, when someone tries to use this approach on me I am inclined to instantly write them off and never look back. I’m not saying everyone is like me but you might want to be wary of what kind of people you are optimizing yourself for.
I read “response” much more broadly than you do. I’d translate the question as something more like, “What sort of currency are you currently lacking?” or “What sort of aid do you currently require?”
Imagine someone says, “Work has gotten really overwhelming.” There are many things this could mean, and many ways you could potentially help them. Perhaps they suspect they are making a strategic error in their work, and you can help by analyzing strategic options with them. Perhaps they are in danger of a bucket error suggesting that they are a bad person for letting work get overwhelming, in which case you can help by providing evidence that you don’t think they’re bad. Perhaps they are tired, and you can help by bringing them to a state that’s more restful. If you don’t know which of these things is more likely, asking is a pretty good shortcut to figuring it out.
I initially read it in the same way you did, however I also think SquirrelInHell has a point. But I would say the place where he’s going wrong is sometimes (possibly most of the time) people don’t know what it is they’re seeking from a conversation. A lot of people don’t know themselves well enough, so having been promted with that question allows them to properly introspect, perhaps.. but I do agree that password based conversations are frustrating.
I would say that even if you’re right, you’re still wrong. Let me explain: My perception is that the advice given in the OP is pretty standard, and I have received it quite often myself; both (1) the idea that sometimes people are looking for advice, and sometimes they are looking for sympathy, and it’s useful to know which is which before responding, and (2) less frequently, the idea that you should explicitly ask them which one.
In light of that, my perception is that you are saying “anybody who follows this script that a huge number of people have been credibly advised to follow, I would write them off and never look back.” That seems like a bad approach even if the script they’re using is bad for some reason, because it’s going to result in writing off huge numbers of people who took reasonable advice, who probably have nothing against you, and who you probably otherwise have nothing against.
More generally, it is my perception, as someone who thinks a lot about communication styles, that there are two opposing schools of thought on this particular issue. If someone tries to reason about how to communicate best, learn the most effective advice and scripts for communicating with people, and try to apply it, the two responses I see from someone who notices that this is happening are (1) to appreciate the extra effort that the person is putting into communicating, even if the result didn’t quite land; or (2) what seems to be yours, which is to treat such an attempt as somehow artifical and offensive, and to choose to be offended by it.
I think that in general, there are too many different communication styles in the world for most possible pairs of people to be likely to communicate successfully by only using their own style / instinct / intuition. I think that trying to learn about scripts that will help you successfully communicate with others is a good thing, which makes successful communication more likely. So I think that people who choose (2) above are mostly dooming themselves to be miserable (unless they stick to communicating with people mostly just like themselves.)