In summary, the conversation covers the topic of using tetrad instead of metric formulations in relativity. While some view tetrad as more powerful and elegant, others argue that familiarity with metric formulations may make it difficult to switch to tetrad. However, tetrad is still relevant in areas such as string theory and should be introduced to those studying general relativity. It is also commonly used in problem-solving, with the help of computer programs.
Personally, I haven't formed an opinion yet. The issue was raised at a recent AAPT Topical Conference on Teaching General Relativity to Undergraduates... but I don't think anything was written down on that issue.
I only covered a little bit of vierbeins in a black holes course, relating to coupling spinors to relativity (positive energy theorem), and while they are extremely useful for that process, I'm not sure if I'd liked to have meet them sooner.
Part of the issue might be a common thing in physics (and any subject), familiarity with one method breeds dislike of other methods even if they are ultimately more powerful and elegant. Having spent time learning about metric formulations of GR, to flick to an entirely new system would be a pain.
Saying that, vierbeins come up in string theory too, since you have to couple spinors to GR again and play an important role in things like the selection of the compactified space due to holonomy so an increased understanding of such material wouldn't exactly hurt a great number of theoretical physicists. How relevant GR researchers think string theory is is another question entirely though.
They are definitely something anyone who does a decent quantity of GR should come across, even if it was as I did, a few lectures on how you'd couple matter to a GR-like theory and the relevence to some well known results. If someone knew they were going to go far into GR or to certain areas of string theory then meeting vierbeins sooner would be a definite plus but it's hard to say when you're only in your 2nd or 3rd year of university what you'll be doing 3 years down the line. Looking back with hindsight is always easier...
I'd say that the tetrad approach is very useful, but more complex mathematically, so its usually introduced after one has become familiar with the metric approach.
I tend to use tetrad approaches (using the Newman-penrose formalism) in GRTensor a lot in working problems, but the computer does all the associated mathematical grunge work for me in that case.