Understanding Gibbs $H$-theorem: where does Jaynes' "blurring" argument come from?

My main concern is: does this blurring/loss of knowledge come from any well-known physical law/principle? For instance, should we link the quantization of the phase space to ΔxΔp≥ℏ2ΔxΔp≥ℏ2, or to some kind of observer effect?

The description in the Wikipedia article is misinformed and misguiding. The blurring, or coarse-graining, is merely one possible formal mechanism to get $H$ decrease in time in the (Boltzmannian) $H$-theorem.

This procedure introduces artifical scale below which details are discarded and thus it has little to do with mechanics where nothing gets discarded or the 2nd law of thermodynamics, where we are not concerned with such details at all (we only use macroscopic state variables).

Also $H$ (a function of probabilities and therefore time) is not related to thermodynamic entropy (function of macroscopic state variables) in any simple way.

In fact there is no blurring of the probability function needed to explain the 2nd law. Jaynes has shown how to derive the entropy formulation of the 2nd law for thermally isolated system without any blurring, only using Hamiltonian mechanics, principle of maximum information entropy and the assumption that results of macroscopic experiments are repeatable.

Let the system evolve from thermodynamically equilibrium state $A$ to thermodynamically equilibrium state $B$. We will describe the whole process also by means of probability density $\rho(t)$, starting at time $t_A$ with initial function $\rho(t_A)$ and ending at time $t_B$. The initial density $\rho(t_A)$ can be anything as long as it respects constraints of the macrostate $A$. The subsequent densities $\rho(t)$ for $t \geq t_A$, however, are entirely determined by the Hamiltonian and the initial condition; we are not free to choose them.

Due to the Liouville theorem, the information entropy (often called Gibbs' entropy)

$$ I[\rho] = \int -\rho\ln\rho \,dqdp $$

remains constant in time, there is no blurring of $\rho$ of any kind . (1)

It will be shown that thermodynamic entropy in the final equilibrium state $S_B$ is greater or equal to the initial thermodynamic entropy $S_A$.

This result is possible because thermodynamic entropy of a macrostate, in general, is not simply proportional to information entropy of the time-dependent probability distribution $\rho(t)$. Its relation to the concept of information entropy is this:

Value of thermodynamic entropy of macrostate $X$ is given by the value of information entropy for that probability distribution $\rho_X$, which is both consistent with macrostate $X$ and gives maximum possible value to the information entropy. (2)

Now, obviously $\rho(t_B)$ is consistent with macrostate $B$ but $I[\rho(t_B)]$ is not necessarily the maximum possible value of $I$ for all $\rho$'s compatible with the macrostate $B$.

The probability density that is not only consistent with macrostate $B$ but also maximizes information entropy is, in general, different from $\rho(t_B)$. Let us denote this maximizing density as $\rho_B$; then the relation of the two information entropies is

$$ I[\rho_B] \geq I[\rho(t_B)]. $$

Now, based on (2) we can write this as $$ S_B \geq I[\rho(t_B)], $$ that is, thermodynamic entropy in the final equilibrium state is higher or equal to information entropy of the evolved probability distribution.

Based on (1) we can write this also as $$ S_B \geq I[\rho(t_A)]. $$

This means that whatever the density $\rho(t_A)$ is chosen to describe the macrostate $A$, thermodynamic entropy of the final state $B$ is equal or higher than information entropy of $\rho(t_A)$. And so if the density $\rho_A$ is chosen such that it maximizes $I$ under constraints of the initial macrostate $A$, $I$ attains value of thermodynamic entropy $S_A$ and we obtain the inequality

$$ S_B \geq S_A. $$

This means we derived entropy formulation of 2nd law of thermodynamics from the principle of maximum information entropy, using constancy of Gibbs' entropy as one of the ingredients.

More generally, is quantum mechanics needed (as suggested by this answer to a related question) to explain completely how the irreversible dynamics observed in nature at the macroscopic scale emerge from reversible laws?

If we wanted to explain the process completely down to elementary particles, a theory of these particles such as quantum theory would be needed. But if the goal is merely to show that irreversible evolution of macroscopic variables is consistent with reversible evolution of microscopic variables, then the answer is no.