How can non-parametric bootstrapping approximate a sample distribution given the central limit theorem?

I'm taking an advanced statistics module as part of my graduate course. While at undergrad we were taught to check the assumptions of our data via tests like Shapiro-Wilk, we're now encouraged to do this using non-parametric bootstraps. My confusion starts because the central limit theorem (CLT) says that, over enough samples, any distribution of a statistic should approach a normal distribution. Let's say I want to test if the mean of some variable ( $\bar{x}$ ) in my sample is significantly different from what I expect ( $\bar{x} \ne \mu$ ). I perform a non-parametric bootstrap for the distribution of $\bar{x}$ across theoretical samples, which I will use to decide whether to perform a parametric t-test or a non-parametric Wilcoxon. But if CLT is true then the distribution of $\bar{x}$ will be normally distributed regardless of how the original sample's distribution of $x$ is shaped. How then can I make infer anything about my sample's distribution or the test I should use if the sampling distribution will always appear normal?