Why not suggest that society and the individual have a joint property right in a product produced by the individual, and that there is an implicit agreement between the individual and society that the allocation of ownership will be done in a fair and reasonable way, as determined by society with some room for flexibility, but not in a blatantly unreasonable way. The reason society also has a partial property right in the product is that it was created by virtue of society—society laid the necessary infrastructure for the product to be produced (roads, healthcare system, sewage, security, etc.).

Are you arguing that the concept of ownership is not defined by society, but rather that the individual gives society the right to define it? But what if there is an individual who does not give that right? What would you say to him? It seems to me that this model is problematic.

I am arguing that the concept of ownership is not defined by society but by natural morality. Just as my right to life is not defined by society, so too my property right in a product that I created is not defined by society (self-evident). But at the same time, one must recognize the fact that when I create a product, I am not its only creator; in fact society is a silent partner in the creation, in that it provides various necessary infrastructures for producing products (roads, sewage, garbage removal, healthcare, education, security, etc.). So in practice, when I create a product like a chair, for example, it has two creators: me and society. The percentages of ownership are not defined rigidly; rather, there is an implicit agreement between the two creators on a fair division within reasonable bounds (a proper purpose is part of what defines the bounds of reasonableness). When the division is made in an unreasonable / improper way, the division is invalid.

There is a difference. The right to life does not relate to others, but ownership by definition does relate to others (it affects others).
Beyond that, one must distinguish between ownership of what you created and ownership of what you acquired. Regarding what you created, there is definitely room for the view that ownership is natural and not a social convention. There are expressions of this among various thinkers, and also in Jewish law. Regarding things you acquired, that is a social act and it requires society’s recognition.
As for your model itself, see Sha’arei Yosher, Gate 5, throughout, for proofs that there is a property system that precedes the command of the Torah and is a product of society’s consent (= the theory of civil law). What you are proposing is a somewhat different version: that society is a natural partner in it, and not that it is constituted by society. Maybe.
One more comment. We also find similar authority in Even HaEzer: “Whoever betroths, betroths subject to the consent of the Rabbis, and the Rabbis dissolved his betrothal from him.” How would you explain that there? In my view, it is the same foundation: personal status and civil law are both the legal part of Jewish law (Even HaEzer and Choshen Mishpat), as in every legal system in the world. And as such, they are entrusted to society and constituted by it. Therefore, the representatives of society have delegated authority in both of these areas. No one would imagine that the sages have such authority in the other areas of Jewish law: Yoreh De’ah or Orach Chayim.

Regarding marital acquisition, there it makes more sense to say that it is entrusted to society and constituted by it, because the husband did not invest effort in order to acquire the woman (as opposed to money), and therefore he has no moral right to exclusivity over her. And if you say that he invested the betrothal money, one can answer that the betrothal money is only a side aspect of betrothal, as evidenced by the fact that one can betroth through intercourse and through a document. The betrothal money is intended only to testify to definite intent, and not as fair payment for the value of the exclusive right to the woman.

As for the distinction between a product I created and a product I acquired, I do not see a substantive difference, because if, say, I built two chairs and traded one of them for a sack of apples, it still makes sense to say that I have a moral right to exclusivity over the apples as well, because I invested effort for them (true, the effort did not involve gardening and watering but rather a different kind of effort, but still effort in quite a similar amount to someone who grew them himself).

The point is not the effort, but the fact that it came from me. See my article on ontological gratitude.

Since the creator of the product has ownership rights over it even without society’s consent, he can permit exclusive use of the product to someone else as well without society’s consent (because he is the owner of the product and therefore decides who is allowed to use it) in exchange for some consideration from the other side. In that way, one can in practice create an effective transfer of ownership even in purchase transactions without society’s consent, drawing its validity from the ontological creator’s right.

I disagree. The fact that you are the owner does not necessarily mean that you can also transfer ownership to others. The transfer and conveyance of ownership may be conditional on society’s recognition of that act, because it is a legal act between two people in society that has implications for society as a whole.
By the way, the king and the religious court can also declare ownerless even property that I created.

What implications are there here for society as a whole? After all, both before the transfer of ownership and after it, society was forbidden to use the object. There is no affecting others here.

It is not “affecting others,” but implications. When the object belongs to the second person, one may not use it.

I mean that from the perspective in which ownership remains with the original creator and he transfers the “right of use” in exchange for some consideration to another party, there are no implications for society, because both before and after the transfer of the “right of use,” society could not use the object.

But when the object is conveyed, ownership does not remain with the first person; it passes to the buyer. And then, as stated, there are implications for society (even if this is not “affecting others”). My claim is that the very fact that the transfer of ownership has significance vis-à-vis society (now use depends on permission from the buyer and not from the original owner; the buyer can transfer it further and betroth a woman with it, or sue someone who uses it without his permission) means that this is done within a framework that society enables and recognizes.