If p-values are random, why not decide using the test statistic instead? I am currently reading the paper

Duncan J Murdoch, Yu-Ling Tsai & James Adcock (2008) P-Values are Random Variables, The American Statistician, 62:3, 242-245, DOI: 10.1198/000313008X332421

In this paper, the authors argue that a p-value is itself a random test statistic. Moreover, given some test statistic $T$ that takes in an i.i.d. sample $\mathbf{X}$ and outputs a real number, a p-value is the probability integral transform of $T(\mathbf{X})$. That is, if $T(\mathbf{X})$ has a cumulative distribution function $F_{\tau}$, then the corresponding p-value is $F_{\tau}(T(\mathbf{X}))$. We can then decide whether to reject a null hypothesis based on this p-value. For example, a decision rule could be to reject the null hypothesis if the corresponding p-value is less than 0.05.
However, because $F_{\tau}$ is monotone increasing, I am not sure why we need to compute a p-value in the first place to decide whether or not to reject a null hypothesis. Can't the value of the test statistic $T(\mathbf{X})$ be used to decide? For example, if the decision rule is to reject the null hypothesis when the p-value is less than 0.05, then, if the inverse of $F_{\tau}$ exists, we can obtain the threshold value for $T(\mathbf{X})$ beyond which the null hypothesis is rejected. Furthermore, we should be able to compute type I and II error rates just using this threshold value for $T(\mathbf{X})$.

$p$-value is a random variable like the test statistic is a random variable, so it's not clear to me what it has to do with the rest of your question. Both are functions of random variables, hence random variables themselves.
Yes, you can use the threshold for the test statistic, but how would you choose it? How would you say that the threshold should be 1.96, 5, or 100? With $p$-values you have a familiar and easily interpretable scale of probabilities, that is not necessarily the case for test statistics.
