Surely someone has mulled over why the universe might exhibit such a non-intuitive and thus interesting asymmetry?
As always, when it comes to valid and important physical theories, the reason why the Universe has non-intuitive features is simply that the intuition is wrong. Arguments based on wrong intuition are irrational and unscientific.
Rationally speaking, there doesn't exist any reason why the laws of Nature should be left-right-symmetric. In particular, spin-1/2 fermionic fields may naturally come in elementary pieces which are 2-component spinors. 2-component spinors inherently allow a left-right asymmetry and because the left-handed and right-handed components of the Dirac 4-spinors may have different charges under the gauge groups, this asymmetry is translated to the interactions, too.
It would be unusual and somewhat unlikely if all the 2-component spinor fields could be combined into 4-spinors in which the pairs of 2-component spinor fields have the same charges under all the gauge groups. The minimal building blocks are the 2-component spinors and as long as we combine them and their charges in a way that is free of gauge anomalies, it's a valid theory. The left-right-symmetric theories form a small (but not infinitely small) fraction of the allowed theories.
The more right way to formulate your question would be to ask Why have we created the intuition that the world should be left-right-symmetric. The reason is that the world is approximately left-right-symmetric. That's because at long distances i.e. low energies, all the forces – specifically the weak force – that make the physics left-right-asymmetric disappear because they're caused by massive messengers (W-bosons and Z-bosons).
The only unbroken group at low energies – and therefore the only long-range force (except for gravity) – is electromagnetism and its $U(1)$ group. Under that group, there is a pairing of the 2-component spinors and physics is left-right-symmetric. This is no accident. The electromagnetic $U(1)$, the unbroken group, is defined as the group under which the vacuum condensate is neutral. Because it's neutral but it's still able to convert the two 2-component spinors in the massive Dirac 4-component spinors into each other (it's what the mass term does), it implies that these two 2-component spinors must have the same charges under this electromagnetic $U(1)$.
The previous sentence doesn't hold for neutrinos because they're probably strictly neutral particles described by Majorana fermions whose "number of degrees of freedom" is exactly the same as for a 2-component Weyl fermion.
One may also obtain left-right-symmetric models by breaking a symmetry in a more fundamental left-right-symmetric model. However, there's really no known valid argument indicating that the starting point should be left-right-symmetric.
In fact, I updated the answer by adding this paragraph. There exists an anthropic reason why the world should better violate similar discrete symmetries. The laws of Nature allows both matter and antimatter. Their properties are always related by the CPT symmetry. But if there were also an exact C symmetry or CP symmetry, the matter and antimatter in the early Universe would be balanced and they would pretty much exactly annihilate with each other. Because some matter is left, the symmetry couldn't be exact. Andrei Sakharov actually formulated the necessary Sakharov conditions for baryogenesis. $B$, $C+CP$ violation, and interactions away from thermal equilibrium. I realize that $P$ violation isn't really there but it's pretty close to $C$ violation which is found in the list. The list is enough to say that violations of discrete symmetries are necessary for life to exist.