Actually, Benford's Law is an incredibly powerful method. This is because the Benford's frequency distribution of first digit is applicable to all sorts of data set that occur in the real or natural world.
You are right that you can use Benford's Law in only certain circumstances. You say that the data has to have a uniform log distribution. Technically, this is absolutely correct. But, you could describe the requirement in a much simpler and lenient way. All you need is that the data set range crosses at least one order of magnitude. Let's say from 1 to 9 or 10 to 99 or 100 to 999. If it crosses two orders of magnitude, you are in business. And, Benford's Law should be pretty helpful.
The beauty of Benford's Law is that it helps you narrow your investigation really quickly on the needle(s) within the hay stack of data. You look for the anomalies whereby the frequency of first digit is much different than Benford frequencies. Once you notice that there are two many 6s, you then use Benford's Law to focus on just the 6s; but, you take it now to the first two digits (60, 61, 62, 63, etc...). Now, maybe you find out there are a lot more 63s then what Benford suggest (you would do that by calculating Benford's frequency: log(1+1/63) that gives you a value close to 0%). So, you use Benford to the first three digits. By the time you find out there are way too many 632s (or whatever by calculating Benford's frequency: log (1+1/632)) than expected you are probably on to something. Not all anomalies are frauds. But, most frauds are anomalies.
If the data set that Marc Hauser manipulated are natural unconstrained data with a related range that was wide enough, then Benford's Law would be a pretty good diagnostic tool. I am sure there are other good diagnostic tools also detecting unlikely patterns and by combining them with Benford's Law you could most probably have investigated the Marc Hauser affair effectively (taking into consideration the mentioned data requirement of Benford's Law).
I explain Benford's Law a bit more in this short presentation that you can see here: