What could be contained in this question, and manner in which we answer it, is going to depend on several things. First of all, we should clarify what we're asking about: your specific question mentioned only phrases that needed 'reinterpretation', but there are many more cases that deserve inquiry, such as gezairah shavas (see the Rambam's introduction to the Mishna, where he breaks it up into 5 components, and we should really ask about the requirement for all 5). Another matter to take into account is the question of how we view the roles of/relationship between 'pshat' and 'derash' - which, in turn, might differ between narrative passages (parts of the Torah that tell a story) and instructive (as in, commandments) sections. The examples in your question contained a sample of each. Additionally, this question can be asked in two forms: 'why is there a clear need for an Oral Law (meaning, why did the Written Law have to be written with contradictions and ambiguities)', or, 'why does the halakha follow something other than the plain meaning', which is asked here, which is a distinct question and not directly meant to be answered here (though I think it is indirectly).
1. Idiomatic Expressions
Some differences between the Written Law and its explanation can be simply explained by saying that the expression in the text of the Torah literally means what the Oral Law says, and that's how it would have been interpreted by any reader of the Torah around the time and place where it was written - even without the Oral tradition. In other words, these are idioms or expressions whose meaning would have been clear to the original recipients of the Torah and we only needed to preserve these peirushim hamekubalim misinai because these expressions might fall out of use and need to be understood thousands of years later. For example, the Torah commands that [these things/words should be] 'totafos' between your eyes (Devarim 6:4). Disregarding the meaning of totafos, why would the Torah say 'between your eyes' when it means 'above your hairline' (Shulchan Aruch 27:9)? The answer, as evidenced by Ugaritic texts, is this is what the phrase actually means, as it is an idiom for the front of the head, not the nose-bridge between the eyes. [footnote 1] Similarly, Rashi's explanation of your example in Devarim 22 indicates that, in fact, a reader of the text without an Oral tradition would actually get it right - if they were a reader in Israel, c. 1200 BCE. (I admit that in this particular case, such an explanation may be unsatisfying). I am fairly certain that the question of Yishmael's 'playing' should be understood in the same way, namely, that an ancient reader would not have been puzzled by this use of the word 'metzchek'. [footnote 2]
2. Levels of Meaning
Let us grant, however, that there are many more cases where peirushim hamekubalim misinai are not idiomatic expressions, but deviations or reinterpretations of the text. The most striking and oft-discussed example is the requirement for an injurer to pay 'an eye for an eye' (Shemos 21:23), which the Sages interpret to mean monetary payment (Bava Kamma 83b). While one may argue that this runs against the plain meaning of the Written Law, it isn't so clear, because the Torah also says (assuming that the Torah is one unified whole, written by One divine Author) "you cannot accept money as reimbursement for a murder" (Bamidbar 35:31), clearly implying that money can be accepted as payment for injury. (However, the Ibn Ezra seems to think that this too is an an idiom, as in the example above).
The first approach is that the somewhat misleading message that is gleaned from seeing only one fragment of the Torah's larger picture is that the Torah intends to convey multiple levels of meaning, which are true on varying level, even if only one of them can be acted upon in terms of halakhic implementation. This seems to be brought out often in the commentary of the Ramban, who often gives explanations of the verses according to Chazal, and then a different explanation 'al pi peshat', and a third 'al derech ha'emes', and the Ramban sees no contradiction; all these meanings are intended by the verse even if only one of them can be used in halakha. In the example discussed, there may be levels in which the strict 'eye for an eye' is indeed true. The Rikanti on the pasuk in Shemos 21:23 asks, why couldn't the Torah just say 'pay money'? He shows that the manner in which the pasuk is written does teach certain kabbalistic truths. The Rambam too, may not think that there are mystical lessons taught by the 'peshat', but the moral lessons to be gained from the mitzvos are better gained from reading the peshat (Moreh Nevuchim 3:41 and in Hil. Chovel U'Mazik 1:1). Thus, the 'partial picture' of reading one pasuk in Shemos actually teaches us Torah messages, if not the halakha. [footnote 3]
3. Emphasizes the Importance of an Oral Tradition
R. Chasdai Crescas in the beginning of Ohr Hashem explains that the reason why we have 'two Torahs' and that the Written Torah is so unclear is so that it be obvious that we shouldn't rely solely on the Written Law. As Hillel told a convert, we need to rely on the Sages and the tradition in order to understand what any word may mean (Shabbos 31a), and besides, there are simply too many details to be fully contained in a book. Therefore, to make sure that we would never make the mistake of attempting to interpret the Torah without the Oral Tradition, Hashem made sure that there would be contradictions, ambiguities, etc. This explanation, in different forms, has been repeated by many others over the generations, including R. Crescas' student, R. Yosef Albo (Sefer HaIkkarim 2:23)
In what may be deemed a variation of this explanation, R. Yaakov Zvi Mecklenberg, in the intro to Hakesav Vehakabalah, says that the Oral Torah is like the 'soul' of a physical body, and is therefore too sublime and extensive to be contained in a physical manner, but rather is left to be recognized as infinite and unending. In other words, the Torah contains so-to-speak 'imprecise' language to indicate that its true meaning is more exalted than anything that can be contained in language.
4. Flexibility of Interpretation
Thus far, we have only discussed explanations of phrases where the halakha may be different that the phrase's plain meaning, but there are other parts to the Oral Law. In the Rambam's division of the Oral Torah into five parts, he names the third part 'hekeshos', things learned from derashot such as 'gezairah shava' and the like. In these areas, it's unclear whether or not they can go against the plain meaning of the text. In Bava Kama 7b it appears that a gezaira shava cannot override the plain meaning of the text, though to what extent may be subject to an argument between R. Akiva and R. Yishmael.
In this area of the Oral Law, which involves the usage of hermeneutical principles, there is considerable amount of flexibility of interpretation, which is why the Talmud is so full of arguments and different opinions. In fact, while many are aware of the concept that one court cannot annul the decrees of an earlier court unless it is greater בחכמה ובמנין (in wisdom and number), the Rambam (Hil. Mamrim 1:1-2) believes that this is only true regarding Rabbinic enactments, but with regards to interpreting the Torah - any Beis Din Hagadol can undo the interpretations of that of an earlier generation. Regarding this specific form of "Oral Law" therefore, some have suggested that the reason for its not being set in writing was in order to allow for flexibility. R. Moshe Shmuel Glasner, in his introduction to Dor Revii, says that the Torah allowed for this flexibility in order to 'match with the times' so to speak, and such an idea seems to exist in the writings of Rav Kook as well. [footnote 4]
Further Reading: David Henshke, אין מקרא יוצא מידי פשוטו. המעין יז,ג (תשלז) 7-19; יז,ד: 52-69
See H L Ginsberg, The Ugarit Texts (Jerusalem Bialik Foundation, 1936) 73, ANET 30 no. 10, and the sources quoted in J H Tigay, "On the meaning of t(w)tpt", Journal of Biblical Literature 101 no 3 S 1982, p 326 n. 30
See Joseph Roth-Rotem, "The Exposition of the Banishment of Ishma'el Story" (Genesis 21:9-21), Beit Mikra: Journal for the Study of the Bible and Its World vol. 43 (1998), pp. 113-125 and M Moreshet, "צחק — שחק; יצחק — ישחק", Beit Mikra: Journal for the Study of the Bible and Its World vol. 13 (1978), pp. 127-130
This explanation only seems viable according to those who differentiate between peshat and derash, and assume that the halakha follows only the derash, as does the Rashbam (Beraishis 37:1), though the opinions of the Ibn Ezra and Rambam may be more complicated, as well as the question of how this relates to אין מקרא יוצא מידי פשוטו. See here for a discussion, and here for more sources. In addition, it is difficult to see how this applies to narrative sections; see "What Actually Happened" question.
Sources quoted here, footnote 5: Ma'amarei ha-Re'i'ah: Qovetz Maamarim, Jerusalem, 1983/4, pp. 1-9 and pp. 113-115, and Orot, Jerusalem, 1981/2, pp. 102-118 and pp. 120-121