Bear in mind some general deficiencies of hypothesis tests of "nowhere dense" sets
To give some context to the many intelligent answers I expect this question will attract, I'm going to start by pointing out a big disadvantage of all tests of this general class, including the KS test, the AD test, and many other variants that test whether or not data comes from a specific distribution or parametric distributional family. These are all tests of a null hypothesis that is "nowhere dense", meaning that we are testing an extremely specific class of distributions in the null hypothesis compared to much larger class of distributions in the altnerative hypothesis. In most cases where we have data from a process, there may be some theoretical reason to think that a particular distributional form might hold approximately, but it is extremely rare to have reason to believe that a narrow distributional form would hold exactly. Even if we have good theoretical reason to think that a certain distributional form would hold (e.g., mean of IID data leading to a normal distribution via the CLT) the assumptions made in the theoretical derivation are usually only approximations to reality, and so the exact distribution at issue is usually still slightly different to the theoretical distribution. Typically this means that the null hypothesis is "always false" and so the test effectively becomes one where the p-value will converge to zero with enough data.
This is the primary problem that motivated the ASA statement on p-value and the general aversion that the statistics community has to classical hypothesis tests with a point-null hypothesis. Many statisticians are averse to this type of test because it is a test of a hypothesis that is so specific that it is a priori impossible, and so if the null hypothesis is not rejected, that is only because we lack enough data for adequate test power. For this reason, many statisticians prefer to use interval estimation methods like confidence intervals and credibility intervals that give an estimate for a range of values of the unknown quantity/parameter/distribution of interest, rather than testing against a specific case. The KS test, AD test, etc., are not easily emenable to conversion to a confidence interval over the function space of possible distribution functions. It is possible to get pointwise confidence bands using reasoning analogous to the KS test, but these usually only give confidence intervals for specific points, rather than confidence sets over the space of distribution functions.
You can of course draw distinctions between distributional tests of this general class, in terms of their power function against specific kinds of alternatives. As you've pointed out in your question, some of the tests are more powerful against certain kinds of alternatives than others, and it is possible to do a deep dive into this by comparing alternative tests with simulation analysis, etc. While this is possible ---and it is useful in understanding the relative merits of all these tests--- this does not get past the primary problem that occurs when we test a point-null hypothesis or a "nowhere dense" hypothesis in a large space. If we test a hypothesis that is so specific that it is "always false" then the test operates as it should, rejecting the null with enough data. The test therefore becomes primarily a test of how much data we have, which we already know.
It is worth noting that it is possible to amend any test of a point-null hypothesis or a "nowhere dense" hypothesis by imposing a non-zero "tolerance" for deviation from the stipulated class within the null hypothesis and amending the test statistic accordingly. With a bit of work the KS test can be amended in this way, as can the AD test and other distributional tests. This solves the problem of testing against a "nowhere dense" region (and is how I would recommend dealing with these types of tests) but it then means that there is some additional arbitrarity in how large you make your "tolerance" in the null hypothesis. Confidence intervals and other region-based estimators sidestep this deficiency in the first place by looking for a region-based estimator of the unknown object of interest rather than looking at the level of evidence of deviation of a stipulated set of values of that object.