Your choice for completing the example sentence raises two questions: the ellipsis of the second as in the construction
rather as x than [as] y
and the use of the definite article the.
When two nouns/noun phrases being compared are in strict parallel, it is common to omit the second comparative term as. This is often stylistically preferable because of its economy: the parallel can be read as a single phrase instead of two.
It is possible that Frank Zappa will be remembered as a composer more than a rock musician...
It is one of the quirks of romance that Johnnie Armstrong should be remembered as a hero rather than a vicious gangster.
For the rest of their lives they are remembered as pigs rather than world respected leaders in their various professions.
An author may, however, include the second as, in this case, for added emphasis:
Their writers are known rather as authors than as travellers. But such books are, relatively, few. Most writers on travel are remembered as travellers rather than as authors, and the value of their works lies not so much in revealing the personality and literary power of the writer as in successfully describing his journeys and discoveries.
By the placement of rather before as authors rather than before rather (like in this clause) and by repeating the comparison in reverse, the author of this sentence is heightening the contrast between the few travel writers known as literary figures and the majority who are known as travellers. Including the second as is one more technique to achieve the same goal.
Or it may be added for clarity, or simply as a stylistic choice:
Ultimately, the conflict provided a military victory for Israel, but it is remembered as “the earthquake” or “the blunder” rather than as a successful defence of both Israel proper and the Occupied Territories.
One reason the author of this last sentence may have chosen to insert the second as is not so much the comparison of two things to one, but rather of two nouns specified by the definite article (the earthquake, the blunder) to an indefinite one (a successful defence). While this is not the strictest of parallels, such a construction is by no means a rule.
I suspect your discomfort arises with referring to an author as an author of his books. Since only one person could have authored his books — you're talking about books he wrote, not some on a shelf behind him — then the definite article is expected.
- Tom Clancy was an author.
- Tom Clancy was an/the author of several bestsellers.
- Tom Clancy was the author of The Hunt for Red October, Patriot Games, Clear and Present Danger, and The Sum of All Fears, all adapted into successful films.
In (1), the definite article is not an option; in (2) either article works, but in (3) only the definite article is permitted, since the books are specific, i.e., his books.
Whoever composed your example sentence may have gotten carried away by the parallel structure or missed how adding of his books would require a change of article.
The most idiomatic way of rendering your sentence is:
The writer once told his family that he would rather be remembered as a teacher of the illiterate than [as] the author of his books.
Since the meaning is perfectly clear without it, the second as may be omitted as a stylistic choice or it may be included for the same reason. If you sense that rather a B than the C violates parallel structure, by all means include it. The definite article, however, is required for the author of his books.