Answer

This is a complex topic, and I cannot cover it fully here. In Sha'arei Yosher, gates 3–4, it is discussed at great length. I will address a few points briefly.
First, I completely agree that presumption of possession is not relevant. On the contrary, in general legal systems it is accepted that in monetary law the evidentiary threshold should be lower, since 51% is enough to tip the scale in favor of one side (if we set aside presumptive possession). The explanation is simple: if we do not do this, it turns out that the winner is the one for whom there is only 49% support, which is even less likely. By contrast, in criminal law (punishment, such as the death penalty), proof beyond a reasonable doubt is required, because there is no second party standing opposite the defendant who would prevail in court, as happens in monetary cases.
But the assumptions themselves—that we follow the majority in capital cases, and that we do not follow the majority in monetary cases—are not simple at all. Neither of them is.
Rabbi Shimon Shkop shows that there are contexts in capital law where we do not follow the majority. For example, it is clear that to convict on the basis of majority alone is not enough, since two witnesses are required.
As for monetary law, on the straightforward reading Tosafot dispute this in Sanhedrin 3 and Bava Kamma 27. Both deal with the contradiction between the two passages (since in a religious court we do follow the majority in monetary cases). Some medieval authorities explained that the religious court case is the exceptional one, and in truth we generally do not follow the majority in monetary matters. Other medieval authorities explained that specifically the case of the majority for plowing is the exceptional one, and generally we do follow the majority in monetary matters. Personally, I tend toward the second approach, and the explanation for the uniqueness of the majority-for-plowing case is given by Rabbi Shimon Shkop (based on Nachmanides regarding engagement gifts). His claim is that if a person makes a claim about the way he chose to act, that claim cannot be rejected on the basis of majority, since he has the right to say that he chose to belong to the minority. So, for example, even if most people in my city eat non-kosher food, I can claim that I ate kosher food, because that is what I chose. What depends on my choice and is not a natural process is not determined by majority. So too regarding a person who claims that he sold his ox for slaughter and not for plowing, even though most oxen in the market are sold for plowing.