We do know that in reality Jews were diverse historically as they are now.
If you mean diverse in terms of their beliefs you have to be more specific. For instance, there is no evidence anywhere that Jews of long ago believed like the Reform movement of 200 years ago that the written Torah was not authentic. (Even Sadducees and Karaites accepted the written Torah as authentic.) So if you're more specific I will better understand what you are trying to say.
Any Jew accepts that we had a biblical period were a Priestly caste performed animal sacrifices in one particular Temple. That is not done now. The Siddur you pray from is only a few centuries old, created to met an historical need for Jews.
The main prayer in the Siddur -- the shemoneh esrei -- as well as numerous other parts (Shema, Pesukei Dezimra, etc.), actually goes back at least to the early days of the Second Commonwealth, when the Second Temple was first put up.
The answer that all new things are only the codification of oral law is very convenient and logical.
It's also true.
Is there any proof? Does the Torah make any reference of G-d giving anyone oral law? Or is this the codification of tradition?
Yes, for example: "There are countless terms in the scripture which are undefined, for example the term "work" in sabbatical law, or the term "slaughtering" is kashrut law. These are all terms which the Torah uses, but does not define. It does not define what "work" is forbidden or how animals are to be "slaughtered."
"There are basic legal concepts and institutions, for existence of which is assumed in the Torah, but which are not further explained. For example, without previously specifying the formalities of marriage and divorce, the Torah states that a first husband cannot re-marry the wife he has divorced, if in the meantime she has been married to another man (Deut 12:21)....
"The Mishnah too yields some evidence of the existence of an Oral Law at the very dawn of Jewish history. There are laws contained in Mishnayot which have relevance only to a state of affairs whcih prevailed before the Jews first settled down in the Land of Israel....
"One Mishnah (Negaim 7:1) deals with leprosy signs which had already appeared upon one of the people before the Torah was given at Sinai. It rules that such signs did not render such a person unclean, even after the Torah was given.
"Another Mishnah (Makkot 2:4) deals with the cities which were set asdie as a refuge for a man who killed by accident. There were three such cities east of the Jordan, and three in the Land of Israel. The Mishnah rules that when the Jews first conquered the land east of the Jordan, though they appointed cities of refuge there, these were not operative until all six cities had been designated. There must have already been an Oral Law in existence at the time of Sinai and when the Jews reached the Jordan: to argue otherwise would mean that these laws were formulated at a time when they no longer applied in practice...."
Oral Law, by H.C. Schimmel (Feldheim), pp. 22-23:
"Indubitably, an unwritten, orally transmitted body of customary law or ancestral traditions circulated in ancient Israel, and it regulated vast areas in human relationships. The legal collections found in the Torah... are to be looked upon as records of ammendments, supplements, or annulments of an already existing body of practice that had long governed the lives of the Israelite tribes."
Exploring Exodus by Nahum Sarna, p 171.
Here's some other examples: "In the seventh month, on the first day of the month, you shall have a holy convocation, all work you shall not do; a day of blowing it shall be for you" [Numbers 29:1]. Is it not ridiculous to think that every year the people would be required to proclaim a holy day, abstain from work, bring offerings and blow the shofar without knowing the meaning of this day, without knowing why they were doing it?
"You shall afflict yourselves" (Leviticus 16:31) on Yom Kippur without explaining what form the affliction should take: standing in the hot sun, suffering lack of sleep, or some other discomfort.
"Whoever does work on the Sabbath day shall be put to death" (Ex. 31:15). Does this include one who carries a heavey Torah scroll up a flight of stairs? Does it include purchasing a stick of bubblegum. Since life and death is involved how can one honestly think this law was not given with an oral explanation?
"An ox or a sheep: him and his son you shall not slaughter on one day" (Leviticus 22:28). This one day is not explained: Does it mean the ordinary day in which the day follows the night, as it written concerning Yom Kippur "From even to eve you shall keep your Sabbath [Lev. 23:32]; or does it mean the day of the offerings in which the night follow the day (ibid 7:15)?
Furthermore, there's solid archaeological evidence that even the most minute details of an oral law have been faithfully preserved. For example, the oral law contains perhaps some 5,000 detailed laws that determine if tefillin are kosher or not. Yet, the tefillin they found in Masada and other places are identical down to the smallest detail. There are other examples. But the point is that if such concrete expressions of Jewish law are the same today -- after 2,000 years of bitter exile, dispersion and persecution -- then that is a good indicator they and the rest of the oral tradition is identical to what was practiced in Israel the previous 1,000 years of sovereignty in the Land.