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.