In short, not very, but it has been influential in developing other concepts. When I was doing my MA in ethnomusicology, we covered it as one of the many systematic frameworks that came in and out of the picture. The problem with it in part was the purpose: Lomax wanted to prove that societies with more restricted sexual and gender roles were more "restricted" in their singing, and that more highly structured societies were more structured in their singing.

First of all, this suggests that each society has their one type of singing which can be measured, rather than a huge ranges of styles and subcultures. Lomax was famous for going through the rural South in the US and field recording folk music, but what's almost as important as the stuff he did record is the stuff he didn't: the popular music of the day. It's like if you went down to Alabama and recorded people playing music, but took out any country music from the past 60 years. Wouldn't be very representative, right? But Lomax saw all of that as "inauthentic," and thought that the "true" music of the South was the "folk tradition." As I explain in another comment, often the "folk tradition" is just the popular music of the past that's been held onto much longer than other areas.

So this introduces a "no true Scotsman" sort of problem, where you can pick and choose which musics are "authentic" and which are not, at least in part to further your hypothesis. And this brings us to the second issue, which is that everything is subjective. How do you make a scale of nasality in the voice, for example? The observer would be quantifying things that are essentially matters of opinion. And since that opinion was often a white, western opinion, the results end up skewed towards that viewpoint.

There have been attempts to revisit Lomax's data and revise the cantometric system, but by and large in ethnomusicology it's treated as an interesting concept with limited utility.

This is a nice breakdown of some of the criticism directed towards cantometrics. It also gives some of the ways in which cantometrics or similar systems have found use, including in Pandora's categorization algorithm that helps find songs listeners will like given certain parameters. We don't exactly know what Pandora uses since it's proprietary, but it's certainly an interesting application of ethnomusicological principles.