Thank you so much for the quick reply! A couple of stupid questions: 1. Assuming that flat sections are the same as horizontal sections, did you mean $\nabla P = 0$ instead of ${\nabla}_{\partial} P = 0$? If something is in the kernel of $\nabla$, then it is definitely in the kernel of ${\nabla}_{\partial}$, but is the converse true? 2. Having zero $p$-curvature means having "enough" horizontal sections. So is the "main idea" behind the construction of $P$ simply to provide sufficiently many horizontal sections? 3. What exactly do you mean by evaluating the section $e$ at 0?

@ulrich Thank you for the reply. I am sorry but I am confused. 1. Is it absolutely necessary that the coordinate ring of $X'$, call it $A$, will be a Cohen-Macaulay (CM) ring? 2. Assume that $A$ is CM. That means that it will be a CM module over itself. Does that mean that it will be a CM module over the $p$-adic integers? We require flatness over $\mathbb{Z}_p$. 3. I don’t see why it is necessary for $A$ to be finite as a module over $\mathbb{Z}_p$. Is finiteness not required to conclude that a CM module has no embedded primes? See [link] (stacks.math.columbia.edu/tag/0BUS).

@MartinBrandenburg To be honest, I have not. I was expecting this to work even in the non-smooth case. Do you think that it fails to be flat for non-smooth complete intersections in general?