Intuition behind conditioning Y on X in the front-door adjustment formula I am trying to understand the practicalities of Pearl's front-door criteria to estimate causal effects under an unobserved unconfounder. First, to give context to the question we have a graph that looks like this:

Where $U$ is unobserved and, hence, no back-door adjustment formula can be used.
We can infer the causal effects on $Y$ of an intervention $do(X=x)$ by using the front-door adjustment formula Pearl (2009 p.81-83):
$$P(y | \textit{do}(X = x)) = \sum_z P(z | x) \sum_{x'} P(y|x', z)P(x').$$
My question is:
What is the intuition of conditioning $Y$ on $x'$ ($P(y|x', z)$) if $Y$ is conditionally independent of $X$ given $Z$? Does that mean, in practical terms, that the way to estimate the front-door adjustment is by using the following:
$$P(y | \textit{do}(X = x)) = \sum_z P(z | x) \sum_{x'} P(y|z)P(x').$$
 A: Y is independent of X given Z, but Z isn't independent of X. Which you're manipulating (or, well, simulating the manipulation, anyway).
An intervention on the value of X changes the probabilities of Zs you're going to observe. That's why we need the first part, summing/integrating over the post-intervention distribution $p(z|x)$.
However, both X and Y are influenced by the unobserved variable 'in the wild'. By intervening on X, we eliminate U's influence on X - but not on Y. It's still there, and we still need to account for it.
The $p(x')$ is the pre-intervention probability on values of X, so you cannot use your final equation. The term is actually derived using the independence assumptions to substitute for a $\sum_Up(y|z,u)p(u)$ term. Since U is not easily measurable while X' can be, we're inferring U from pre-intervention X.
A: To answer your question straight,
$$P(y | \textit{do}(X = x)) = \sum_z P(z | x) \sum_{x'} P(y|x', z)P(x').$$
can't be rewritten as
$$P(y | \textit{do}(X = x)) = \sum_z P(z | x) \sum_{x'} P(y|z)P(x')$$
because $P(y|x', z) \neq P(y|z)$ in the pre-intervention distribution given the presence of U, the confounder.
