On the derivation of confidence intervals for hypothesis testing I understand that a 95% confidence interval should contain the value of the population parameter under the null hypothesis 95% of the time. That is, for $H_0: \theta = \theta_0$:
$$
\begin{align*}
&\Pr(|t| \leq t_{crit}) = 0.95 \text{ under } H_0.\\
\Rightarrow &\Pr\left(\frac{\hat{\theta} - \theta_0}{se(\hat{\theta})} \leq t_{crit}\right) = 0.95 \text{ under } H_0.\\
\Rightarrow &\Pr\left(\hat{\theta} - t_{crit} se(\hat{\theta}) \leq \theta_0 \leq \hat{\theta} + t_{crit} se(\hat{\theta})\right) = 0.95 \text{ under } H_0.
\end{align*}
$$
That is an interval centered around $\theta_0$. Another way to write it down is:
$$
CI = \left[\hat{\theta} - t_{crit} se(\hat{\theta}), \hat{\theta} + t_{crit} se(\hat{\theta})\right]
$$
We can then reject $H_0$ (at a 5% significance level) if $\theta_0$ is not an element of $CI$. My questions is how can that be the case if by construction, we are centering $CI$ precisely around $\theta_0$.

That isn't quite the case.  We center the confidence interval around the estimate of $\theta_0$, namely $\hat{\theta}$.  So the estimate is always in the CI but it isn't the case that the estimand is in the CI.  Hence, the probabilistic interpretation.
