The native "Es funktioniert nicht" negation doesn't use an auxiliary form; there is no parallel structure there on which to build an intuition, even though English has Germanic roots. It's more of a cognate to "It functions not".
Also it helps to imagine that there is always an auxiliary. If you don't see it, it is invisible, but still there. When the auxiliary is invisible, it cannot take the tense, number and so on; these markers go to the main verb.
Now when you make a question, the auxiliary and sentence subject exchange places (subject-aux inversion). This still happens when the aux: is invisible.
Visible aux (will):
Statement: You will have the soup.
Question: Will you have the soup?
Invisible aux:
Statement: You [invisible] have the time.
Question: [invisible] you have the time?
But now, this invisible aux is dangling up in front of the clause and that is forbidden. In this situation, the invisible aux is instantiated in a visible form, and it appears as the verb "do":
Question: Do you have the time?
And, as usual, this invisible-come-visible aux now takes the tense, number, person, "not" negation away from the main verb. With that we can more or less semi-accurately understand how "do" works in forming questions.
Personally, I go for "It not worksy" (or shorter: "Not worksy!") when I want the language to be as broken as the thing it describes.