Regarding the calculation of $p$ values in one tailed tests

Question

In two tailed hypothesis tests, under symmetric sampling distributions, it is necessary to double the $p$-value, which makes sense from the definition of a $p$-value, given that it is equal to the conditional probability that we obtain test statistic at least as extreme as the one that represents our sample given that $H_{0}$ is true.

Now, from this definition, if we are considering a symmetric sampling distribution, shouldn't we also double the $p$-value under one tailed tests? given that there are both negative and positive test statistics as extreme or more extreme that the observed one

Perhaps this has to do with the fact that in one tailed tests, we have tails with an area of $\alpha$, meanwhile in the two tailed case, we have tails with an area of $\alpha/2$, although I am not sure this is the reason.

Answer 1

No, we are picking one tail. If we double the p-value, then we are doing a two-tailed test. By picking one tail, we are choosing to focus on a particular direction of difference and increasing our power to detect a difference in that direction. The tradeoff is that we sacrifice all power to detect a difference in the other direction.