In normal spoken English one does not speak the names of any punctuation. The intended meaning in spoken language is carried by inflection, tone, pauses and so on.
One use of quotes marks in written English is the so-called "scare quotes". These are used to indicate irony. Example (from wiktionary)
Maybe you should ask your "friend" what happened to the money.
It suggests that the person is not a real friend. The use the word is ironic.
In speech that could be indicated by tone, or by a modifier phrase
You should ask your so-called friend...
Modifiers in English normally go before the noun. One way of indicating irony is to speak the quotation marks, but as a modifier before the noun
You should ask your quote-unquote friend...
Putting quote-unquote together in front of the noun fits better with the normal English pattern of adjective before noun. The expression is not actually indicating punctuation, but has the same function as "so-called"
If a longer phrase is actually being quoted then normally no words are required, the start of the quote and its end are implied by the context.
Finally remember, English is a natural language, and sometimes there is no logical reason for something.