books generally - Why was Torah SheBe'Al Peh not allowed to be written?

The Gemara states that it is forbidden to write down the Oral Torah ("Torah SheBe'Al Peh") [1]:

דרש רבי יהודה בר נחמני מתורגמניה דרבי שמעון בן לקיש, כתיב: +שמות ל"ד+ כתוב לך את הדברים האלה, וכתיב: +שמות ל"ד+ כי ע"פ הדברים האלה, הא כיצד? דברים שבכתב אי אתה רשאי לאומרן על פה, דברים שבעל פה אי אתה רשאי לאומרן בכתב. דבי רבי ישמעאל תנא: אלה - אלה אתה כותב, ואי אתה כותב הלכות.

Why was this the case? Wouldn't it have helped the spread of Torah knowledge if it could be written down?

[1] Gittin 60b: “R. Judah b. Nahmani, the public orator of R. Simeon b. Lakish, discoursed as follows: It is written (Exodus 34), ‘Write thou these words,’ and it is written, ‘For according to the mouth of these words.’ What are we to make of this? — It means: The words which are written down you art not at liberty to say by heart, and the words transmitted orally you are not at liberty to recite from writing. A Tanna of the school of R. Ishmael taught: [It is written] ‘These’: these you may write, but you may not write ‘halakhot.’”

Answer

There are many answers given to this question:

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  The Midrash cited in the other answer refers to the exclusive relationship that the Oral Torah allowed. The Christians may read the written Torah, but only the Jews had the Oral Torah. (The Gemara in Gittin also quotes that Passuk.)

  
  


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  The Rambam in Moreh Nevuchim (I.71) takes this further and says that by keeping the Torah oral, it prevented machloket, for people couldn't misinterpret intervening texts, and the Sanhedrin haGadol ruled on every matter.

  
  


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  The Iggeres of R. Sherira Gaon describes how before the Mishnah was written, people were able to learn in their own style (Translation by R. Nosson Dovid Rabinowich):

  
  

Despite the unanimity among the sages in the underlying principles and teachings, each sage taught his students with whichever order and whichever method he preferred.… Some taught general rules; others added details; and others expanded and offered many, many examples and analogies

Putting things in a text format may confine things to a certain extent, while an Oral discussion can be more lively (and perhaps more flexible).

• Perhaps there's another reason. When nothing else is written down, the entire focus of one's studies always remains on the Torah (or Pentateuch) itself. Students are able to constantly re-discover the sources for laws in the Torah itself. Maybe the study of Torah was less about the spread of information than about this constant connection with the Divine word.