Let me offer an interpretation of this sentence.
The verb of the sentence is "doth not make", the subject is the gerund "reading" and the object is "a writer". So the order is in fact subject-verb-object except that part of the verb ('make') is pushed to the end. This is a figure of speech called hyperbaton, and its purpose is to place the emphasis on that part of the verb rather than on the object of the verb. There is additionally the use of the archaic "doth" for "does", but that is a minor matter.
So the emphasis is "Reading does not make you a writer."
In this particular case it is also an idiom, that is to say, a peculiar arrangement of words that follow special phrasing, different than normal grammatical rules might demand, but commonly used and so commonly accepted.
And, just to add to the mix, there is also an ellipsis in there that is not really obvious. Here the verb "make" is actually trivalent, the subject is "reading" but it has two objects, "you" and "a writer". The first of these is omitted by ellipsis, which de-emphasizes its importance, pushing the emphasis back onto make, which is already emphasized by the hyperbaton.
So this little short phrase has a lot going on. Three figures of speech, idiom, hyperbaton and ellipsis, and an archaic verb particle. Not bad for a six word sentence.
The transition point was about a century ago
Note that if we substitute a pronoun (I, he, they) for "why", the transition point comes much later (1980 for "I") – or hasn't even happened yet (all other pronouns). I can't explain why that is, except by pointing out that this very sentence is an increasingly typical usage. Maybe we all tend to be a bit less formal when introducing our own selves into the text.
It's only my opinion, but I think can't (similarly, let's) are examples of grammaticalisation. The contracted form has effectively taken on a "life of its own", leading to a situation where OP is prepared to accept that there may be contexts where can't is "grammatical", but "cannot" (or the equivalent "can not") wouldn't be valid.
Using cannot / can not might be a bit stilted in many contexts now we're so used to seeing the contracted form, but I don't think it's ever ungrammatical.
Best Answer
You form questions in English by inverting the subject and the verb. For the OP's two sentences, the verbs you use in this inversion are does and doesn't, since you can't separate does and n't.
Historically, this may have developed because in Elizabethan English, both Why does this simple code not work? and Why does not this simple code work? were acceptable word orders. Only the first one is acceptable today.
For an example of both word orders in Elizabethan English, Shakespeare used: