Your question and its formulation are excellent. You deserve a real answer, or at least an honest attempt to provide one. The three examples that you gave are used in different contexts. Each will be explained separately.
With regard to the example of hamotze mchavero alav haraya [Bava Kamma 46a], the Gemara states, the burden of proof rests upon the plaintiff, because he is the party claiming redress. This is similar to one claiming to be ill, having the burden of finding a medical clinic [Rav Ashi, id.]. Every rational legal system recognizes the plaintiff has the burden of proof (as a general rule). Accordingly, Chazal state, “Lamma lee kra? Svora hu.”
Although Torah laws themselves must be stated in order to provide sufficient warning [Sanhedrin 56b] and ample reward [see, eg., Chulin 142a], the underlying logical concepts need not be stated. Sefer HaChinuch views most of the ten commandments and many mishpatim as necessary to prevent destruction of civilization [see, Mitzvos 33-37, 49-54]. These mitzvos, commonly understood as essential, or mitzvos sichlios (rational mitzvos) are nonetheless written in the Torah. The logical concepts on which they are based are not stated and they are not independent mitzvos. Tzlach [Berachos 21a] states that logic is insufficient to create a mitzvah d’Oraisa.
The phrase “lamma lee kra” is common throughout the Gemara when seeking to discover why the Torah states a particular point that otherwise would not have been understood. However, the specific phrase, “Lamma lee kra? Svora hu” seems to be reserved for only three concepts. The first, was your example of hamotze mchavero. The second is mai chazis d’dammach didach sumach t’fei (what makes it appear that your blood is redder) [Sanhedrin 74a; Pesachim 25b; Yoma 82b]. This means, no person has the right to claim that his life is more precious than another’s life. There is an equal right to live. The third is peh sh'asur peh sh'hitir (the mouth appearing to restrict, is the same mouth that permits). This means, one may qualify his own statement, so that it is understood in its entire context [Kesuvos 22a]. Thus, the Gemara stated, “Lamma lee kra? Svora hu” only with regard to the most basic rights: life, freedom of expression and ownership (life, liberty, property).
Regarding mai chazis, logic alone is not the basis of the mitzvah in the Torah not to take another’s life. The logic of mai chazis is dependent upon the commandment of lo seertzach (you shall not murder). Only after the Torah commands us not to murder, is logic able to dictate that murder is prohibited even to preserve one’s own life. Since this prohibition is fundamental to coexistence, logic is merely used to extend lo seertzach to a situation where one’s personal desire to live might cause him to contemplate taking another’s life. However, the exceptions to lo seertzach must be explicitly stated in the Torah. These include, killing a rodef (to stop murder or certain cases of rape)[Sanhedrin 74a, based on Devarim 22:26]; habba b’machteres (for self defense)[Sanhedrin 72a, based on Shemos 22:1]; and Rabbi Akiva’s ruling that one should drink from his own canteen in order to survive, rather than sharing the water and thereby perishing [Bava Metzia 62a, based on Vayikra 25:36].
It might be argued that there is a fourth case mentioning, “Lamma lee kra? Svora...hu.” However, the wording states, “Sevora b’alma hu” [Niddah 25a]. Here, the Gemara uses the usual format of seeking to discover circumstances where a pasuk discloses an aspect of halacha that would not be understood on the basis of logic. (Similar examples include Yevamos 47a; Kiddushin 41b; inter alia).
With regard to your second example, of shlo tehay choteh niskar, this is a basic principle in every society. A criminal should not benefit from the fruit of his crime. This concept is found in Tanach several times. For example, in Malachim I [21:19], “will you murder and also inherit?” Again, this is a principle upon which rulings are made. It is not an independent mitzvah.
A related concept is not permitting one to transgress, in order to prevent another from sinning. This is expressed as, vchee omrim chateh keday sh’yiske chavercha? [Shabbos 4a]. As such, one may not harm himself spiritually in order to benefit another. In this regard, Rambam stated, “I have never heard a more remarkable saying than that of the Rabbis in Talmud Yerushalmi, in the ninth chapter of Nedarim, where they strongly criticize those who obligate themselves by oaths and vows, resulting in being bound like prisoners. The exact words they use are, ‘Rabbi Edy said in the name of Rabbi Yitzchak, “Don’t you think, what the Torah prohibits is sufficient, that you need to take on more prohibitions?” [Shemona Perakim, Chap. 4]. This is a remarkable example of the basic logic of not impairing oneself. Similarly, the Mishnah [Bava Kamma 90b] states that one does not have the right to physically harm himself. Yet, the Torah does not expressly prohibit it as an independent mitzvah.
With regard to your third example, kull d'allim gvar (stronger position prevails) [Bava Basra 34b], this is limited to a financial dispute where, as you stated, there is neither evidence nor presumptive ownership. It applies to circumstances where neither party had prior possession. Therefore, there can be no presumptive ownership. Since evidence is not available, it is wrong for a court to become involved with a matter it cannot resolve. This is the reason the parties are left to resolve their dispute outside of beis din. The logic of kull d'allim gvar is, the resoluteness of a party investing more effort to obtain the item in dispute, indicates he may be right. He may eventually come up with proof, since he is making more of an effort [Rosh, Bava Metzia 2a]. The opposing party would reason, why should I bother, when it is possible my opponent will be able to produce evidence the next day? [Id.] In the interim, kull d'allim gvar has no evidentiary value. It simply recognizes the possibility of someone having legitimate possession of an item that had previously been in dispute. Obviously, the Torah does not need to state various possibilities of what may happen when there is no judicial resolution.
There are other logical concepts that are commonly shared by civilized peoples. For example, the act of snatching is inherently dishonest [Bava Basra 33b-34a; Shevuos 32b] and during a legal proceeding, failure to deny a claim is generally construed as an admission [Yevamos 87b; Bava Metzia 37b]. These are principles that provide a framework for rulings that are based on human experience. Since the Torah deploys the speech of people [Berachos 31b], so we can relate to it, certainly the Torah does not need to tell us basics of human interaction. No system of law attempts to do so, but similarly relies on logic.
