But, is that it? In the US, at least, not a lot of employers will give a straight up test of general mental ability (IQ test), likely because they see it as potentially attracting lawsuits. This is because IQ tests show a well-documented racial disparity in scores, which has not been shown to be correlated with anything genetic. Rather, the problem is either traced back to differences in family socioeconomic status growing up, or cultural bias issues with the tests.
Everything in the previous paragraph can be verified in a few minutes of Googling, so, I'm not going to litter this comment with citations when the obvious keywords will produce the information mentioned here. But, everything else, including what's to follow this, should be considered pure speculation on my part.
Given all this context, I can only think of 2 obvious reasons companies resort to using LC or LC-style interviews:
1. As a proxy for a test of general mental ability, as previously mentioned, or
2. Cargo culting, as with the whole "How would you move Mt. Fuji?" phenomenon.
I don't personally have a great deal of insight into which it is, or, if there's a third explanation that I'm completely missing. I also don't know whether there's any research supporting or refuting the idea that LC puzzles provide a valid interview signal for software engineers. All I really know is that these type of problems exercise skills that are not what one uses on the job as a SWE, so, it seems rather illogical to use them as a main component of your hiring process.