There can be no single state-of-the-art for goodness of fit (for example no UMP test across general alternatives will exist, and really nothing even comes close -- even highly regarded omnibus tests have terrible power in some situations).
In general when selecting a test statistic you choose the kinds of deviation that it's most important to detect and use a test statistic that is good at that job. Some tests do very well at a wide variety of interesting alternatives, making them decent default choices, but that doesn't make them "state of the art".
The Anderson Darling is still very popular, and with good reason. The Cramer-von Mises test is much less used these days (to my surprise because it's usually better than the Kolmogorov-Smirnov, but simpler than the Anderson-Darling -- and often has better power than it on differences "in the middle" of the distribution)
All of these tests suffer from bias against some kinds of alternatives, and it's easy to find cases where the Anderson-Darling does much worse (terribly, really) than the other tests. (As I suggest, it's more 'horses for courses' than one test to rule them all). There's often little consideration given to this issue (what's best at picking up the deviations that matter the most to me?), unfortunately.
You may find some value in some of these posts:
Is Shapiro–Wilk the best normality test? Why might it be better than other tests like Anderson-Darling?
2 Sample Kolmogorov-Smirnov vs. Anderson-Darling vs Cramer-von-Mises (about two-sample tests but many of the statements carry over
Motivation for Kolmogorov distance between distributions (more theoretical discussion but there are several important points about practical implications)
I don't think you'll be able to form a confidence interval for the cdf in the Cramer-von Mises and Anderson Darline statistics, because the criteria are based on all of the deviations rather than just the largest.
The same considerations apply as to the distribution of the Kolmogorov–Smirnov test statistic discussed here. The Anderson–Darling test statistic (for a given sample size) has a distribution that (1) doesn't depend on the null-hypothesis distribution when all parameters are known, & (2) depends only on the functional form of the null-hypothesis distribution when location & scale parameters are estimated. I don't know of an R implementation of the A–D test specifically for the exponential distribution with estimated rate parameter, but you could quickly make a function to calculate the test statistic by adapting the ad.test function from the nortest package: change the distribution function from the best-fit normal, pnorm((x - mean(x))/sd(x)), to the best-fit exponential,pexp(x/mean(x)). Then get critical values for any desired significance level & sample size by simulation.
As to the "best" test, note that different tests are more powerful against different kinds of departure from the null-hypothesis distribution. If you have a quite specific alternative in mind, e.g. a Weibull distribution with shape parameter greater than one, a likelihood ratio test will be more powerful than a general-purpose goodness-of-fit test. For more vaguely specified alternatives it might be helpful to compare the power of various tests against a rogues gallery, following the approach of Stephens (1974), "EDF statistics for goodness of fit and some comparisons", JASA, 69, 347.
Best Answer
The Kolmogorov-Smirnov looks for the largest difference between the cdf and the empirical cdf of the data.
There's another test -- the Cramér-von Mises test -- which looks at the sum of squares of the differences in cdf (at the data). It's often somewhat more sensitive than the Kolmogorov-Smirnov to the kind of differences we tend to want to pick up (because it can "accumulate" small but consistent differences rather than needing a single large one).
The problem with both of those tests is that the tail of the cdf is more precise than the middle (in the same sense that a sample estimate of a population proportion near 0 or 1 has lower variance than one near 0.5).
The algebra to show this is not particularly difficult, but we can observe this quite easily without doing the mathematics; we don't need to perform it to obtain intuition about what's going on; there's something even simpler that we can do.
Here I simulate 100 sets of data drawn from a standard uniform, each data set has a sample size of 25. I then draw the empirical cdf of each one (the first such data set is shown in blue; all the rest are in grey and for those I don't plot the point at the left of each step, just the step itself):
As we see in the plot, the (vertical) spread is widest when the population cdf is close to 0.5 and narrowest when the population cdf is close to 0 or 1; the population cdf ($F$) is a diagonal line from (0,0) to (1,1). This pattern of changing spread in the sampling distribution of the empirical cdf happens for every distribution; the spread (specifically, the standard deviation of $\hat{F}$) relates only to $n$ and $F(1-F)$.
We can make use of this additional information that simple tests like the Kolmogorov-Smirnov and the Cramér-von Mises test ignore.
If you calculate a weighted version of the Cramér-von Mises (with weights inversely proportional to this variance) then you end up with the Anderson-Darling statistic; which is to say, it correctly (optimally in a particular sense) accounts for the fact that the cdf in the tail is more precisely estimated; this makes it more sensitive to the differences in the tail than the first two statistics, which don't use the fact that we can estimate the cdf of the tail more precisely.