Many econometrics textbooks (e.g. Wooldridge, "Econometric analysis...") simply write something similar to: "If the population model is $y = xB + u$ and (1) $\text{Cov}(X,U) = 0$; (2) $X'X$ is full rank, then OLS consistently estimate parameters $B$". I've worked out the math behind consistency but I'm a bit lost in interpreting the meaning.

We're talking about consistent estimation, but estimation of what? Please tell me if the following points are correct:

1. If we have random sample $X,Y$ and $X'X$ is invertible, then we can always define Best Linear Predictor of $y$ given $x$. And then OLS always consistently estimates coefficients of Best Linear Predictor (because in BLP we have $\text{Cov}(u,x)=0$ from the definition). Bottom line: we can always interpret OLS estimates as coefficients of BLP
2. The only question is whether BLP corresponds to conditional expectation $\text{E}(y|x)$. If it does (for which we need $\text{E}(u|x) = 0$), then we can interpret OLS estimates as partial effects.

What is wrong with this or what am I missing? I don't get the point of stating the assumption of $\text{Cov}(u,x) = 0$, if this assumption is by definition fulfilled in case of Best Linear Predictor. And if we want to give structural interpretation to BLP coefficients (partial effects) we need stronger assumption of $\text{E}(u|x) = 0$ anyway. Is there any other interpretation we may have with zero covariance?