The OP references 4 specific arguments against the validity of an oral law as they appear in a linked article. This answer will address the article's points.
1) It's too easy to corrupt- The article claims that since humans tend to bungle perfect communication without writing it down, therefore there can be no authentic preserved oral law. An example is given about a wife who sends her husband to the store. If a list is not written, then wrong items will be purchased or correct items forgotten. This is known as the "broken telephone" argument.
Answer: When one person tells one person something, it can be mistakenly repeated. However, if his wife sent 100 people to the store and asked each of them to buy the very same 5 items, we would easily know who made a mistake as the vast majority of people will return with the correct 5 items. In case of mass communication, an oral law is actually the best verification of facts there could be!
A nation that teaches a set of oral rules to all their children as a religious duty is a mass population communicating to a mass population. Things will tend to be very accurate. Mistaken individuals will obviously be corrected by the masses.
2) The article quotes a verse which occurred slightly before the 10 commandments were given which reports that Moses wrote all the words of G-d. Other supporting verses are brought showing that the Torah was written. So, where is the oral law?
Answer: First of all, that verse appears before the entire Torah was given. Even the article's author must admit that the 10 commandments were not yet written in that verse either! The oral laws came later on Mt. Sinai. during the revelation and the 40 days and nights.
Also, the oral law has to do with correct interpretations. Any book's interpretations, are by definition, not included in its writing. So if I told you that "I wrote all G-d's words down.", that doesn't mean I also wrote all the explanations and clarifications of anything.
3) The oral law is open to corruption by the leaders. So how can we think we now have a true oral law?
Answer: First of all, The OP's question is just as good when aimed at anyone's written laws! People in power can change the books and sell their new copy! Did all ancient populations know how to read?? Sometimes only the priests did. How easy was that to change? When it is a religious duty of the masses to recite and memorize things orally, you can't cook the books so fast.
A society with only a written law is vulnerable to the risk of falsification on some level. A society with only an oral law is also subject to the risk of falsification on some level. However, a society that has both a written and oral law, is highly insulated against falsification. It is a check and balance system that defends itself.
4) The article sites the incident in Tanach where King Josiah "..finds the Scroll of the Law in the Temple..". The claim is that Josiah wrote the Torah, and made it all up. Specifically, he invented the Passover laws.
Answer: The Torah scroll in the story was a special copy of the Torah kept in the Temple. The people under previous Kings Amon and Menashe, fell into idolatry. King Menashe had tried to suppress the correct worship of G-d and did not safeguard ancient scrolls in the Temple properly. King Josiah was happy to "find" the scroll because he didn't think it would be easily found after the previous two kings controlled the Temple.
However, any claim that he invented Passover laws or the idea of a ritual Passover rite which the people never heard of is absolutely absurd.
Joshua 5 and II Chronicles 30 both describe great Passovers in history that the Jews celebrated (under Joshua and King Hezekiah respectively). No one claims that Josiah discovered the "Book of Joshuah" in the Temple. :) King Hezekiah reigned right before his son Menashe and was true to G-d and His Torah. Therefore, the people in the time of King Josiah, knew what Passover was, but they just fell into lax observance during the time of Menashe and his son Amon.
Anyone who draws absurd conclusions from the "found the scroll" story is making a mistake due to lack of scholarship. King Josiah did not live in a vacuum. :)
I hope this helps. :)