The problem is (2), as others have noted. Let $X_1, ..., X_n$ be iid normal with mean $\theta$ and variance $1$. The statistic $T(X_1, ..., X_n) = X_1$ is complete, because the normal family is complete. But it is not uncorrelated with all unbiased estimators of $0$; take $\hat 0 = X_1 - X_2$. In constructing unbiased estimators of $0$ we are allowed to use the entire data, whereas completeness is only a property of the marginal distribution of $T(X_1, ..., X_n)$.
The logic in Casella and Berger is this: if $T = T(X_1, ..., X_n)$ is sufficient then it suffices to only consider the distribution of $T$ when looking for unbiased estimators, by Rao-Blackwell. This is what sufficiency is giving you - it allows you to ignore everything except $T$. So it suffices to show, if $g(T)$ is unbiased for $\theta$, that $g(T)$ is uncorrelated with every unbiased estimate of $0$, $\hat 0(T)$ [note: because of sufficiency, we have reduced the problem of showing uncorrelatedness with every estimator of $0$ to only have to show it for estimators that depend only on $T$]. But if $T$ is complete then there are no unbiased estimators $\hat 0(T)$ other than $0$ which $g(T)$ is uncorrelated with, so we are done. Of course, now that we've established that $g(T)$ is the UMVUE, it follows a posteriori that $g(T)$ is uncorrelated with all unbiased estimators of $0$, $\hat 0(X_1, ..., X_n)$, that depend on the entire sample.