There are a number of advantages and necessities for an Oral Torah. Here are a few of the main reasons.
a) Almost all systems of education rely on oral transmission. For example: Medicine, law, and physical skills. You need a person to teach you diagnosis, practice of law, swimming and other methodology and knowledge. Similarly, how to live according to the Torah and live as a Jews requires human interaction.
b) The Written Torah can remain “portable and compact” with the existence of an Oral Torah. Jews have been expelled from many countries and our books have been burnt, stolen or lost. We have a system where even with a fairly small text, the Five Books of Moses, and the information we carry in our minds, hearts and in our practices (the Oral Law) we can still survive as Jews without our books. Also a Torah in which everything would be written and explained would be huge, expensive and impractical to use and transport.
c) The spoken teaching is harder to steal and distort than written word. For example the punishment of “an eye for an eye” is not exactly literal, as explained by the Talmud. Other cultures and religions have plagiarized and distorted the Written Torah because it is so accessible. This happened to a much lesser degree to the Oral Torah because it is less accessible to the outsider.
d) The “living Torah” and the study ethic. Study of the Oral Law facilitates to incorporate the entire Torah into our being, as the verse states: “Write them (Torah teachings) on the tablets of your heart” (Proverbs 3:3). Having most of the Torah in an oral manner has meant that in order to preserve it the Jews must study it. This has encouraged a fantastic, widespread study ethic, passion for learning, and a tradition of intellectualism among the Jews.
e) Necessity of personal contact with transmitters of tradition. When the Torah is oral, one needs to make contact with a previous generation, hence reducing the generation gap and creating inter-generational bonds.
f) Anti-superficiality. Truth requires effort and a search beneath the surface. The Torah is given in such a way that one will only understand the true meaning after much effort, thought and analysis. These are important skills in life and are encouraged by a text which is almost inaccessible at first glance.
g) Feedback mechanism. When Torah is transmitted person to person, as opposed to by a text, there is the possibility of questions, give and take, argument and feedback.
h) Flexibility. There exists the possibility of “changing enough to stay the same”. If everything is written down it becomes inflexible and concreted. When much is oral, it allows flexibility and the ability to change enough to maintain our traditions.
For further reading I suggest “The Oral Law” by H. C. Schimmel, and “The Infinite Chain: Torah, Mesorah and Man” by Natan Lopez-Cardozo.