I am currently relearning econometrics in more depth than I had before. One thing I am trying to make sense of currently is why it is necessary for the assumption of: $$E(u\mid x)=E(u) $$ to be true (where $u$ is the error term).
Here is how I have tried to reason through it, although I am not sure if this is a good reasoning on why.
Let's say $u$ is somehow correlated with some variable $y$, which $x$ is also correlated with. In this case, $$E(u\mid x) \not= E(u) $$ since for greater $x$ values the expectation of the error would go up or down since it is correlated with $x$ through the $y$ variable. With this being the case, the line of best fit would end up with greater or lesser expected errors as $x$ increases and decreases.
Is this what the zero conditional mean assumption is trying to say, or is there a better reasoning that I'm not hitting on?