The one-sample t-Test/z-test and inference validity The CLT tells us that as we collect the means of different samples, the sampling distribution resembles a normal distribution and this way we can infer with a CI on the sampling distribution, the population mean. Now, this can be done via bootstrapping or manual sampling, in both cases the inference is totally valid.
Now, the one-sample t-test also allows you to infer on the population mean with a confidence interval, but how is it valid if it doesn't sample with replacement (bootstrapping) or with manual sampling; meaning; how is the t-test able to infer without incurring in the required additional sampling process that the CLT needs, stated in the previous paragraph? The same question applies to a proportion z-test on categorical data.

The CLT says the sampling distribution of the sample mean is asymptotically normal.
From a single sample, we can estimate the mean and the standard error.  These are our best guesses for each parameter. Given that each is an unbiased estimate for the mean and the standard deviation of the sampling distribution, we apply them as if they were the truth.
So, the knowledge of the asymptotic behaviour coupled with unbiased (and hopefully high precision) estimates allows us to make the inference.  Inference in the case where we have a null about the mean is even simpler.  We have a null (the mean is 0) and since we know the asymptotic behaviour of the sampling distribution, we can compute p values for seeing test stats at least as large as what we've seen.
