I posted this question without having a strong opinion either way - I still don't really, but will post some thoughts!
The very relevant point has been mentioned that the stated purpose of SE sites is to build a "library of detailed answers to every question" about their respective topics. If the operative word there is 'library', maybe we don't want to open the site to a potentially countless number of questions about a countless number of pieces..?
On the other hand, we could read into that phrasing that we want to allow "every question" (where on-topic). That would be fair enough too - the worldbuilding site, for example, says that it aims to "build a library of detailed answers to every question about worldbuilding" - and that's an unlimited set.
So I think it's up to us what we regard as the long-term goals of the site. And arguably, because we allow analysis questions about (potentially) every song, we've already chosen the broader-minded view - that we're (in theory) happy with a potentially unlimited number of questions.
The point has been well-made that it's not necessarily as easy to search for these questions as some other types. And though this is true, I think there's still a whole range of ways in which people might find these questions...
- Good descriptive titles that include the name of the work in question, and the technique (once we've identified it)
- HNQs and the site homepage, for new questions
- 'similar questions', on the right hand bar
- external links to the SO site, and external searches
- User profiles
So what would be the downsides? I think one problem that Dom pointed out is the site facilities - we don't have native video or sound embedding the way we can currently host pictures, and that's an impediment to asking and (especially) answering. One can imagine that with a site that can use your webcam to take a quick video on request, many questions could be answered easily.
Ultimately I think the most important thing would be the balance of the number of people interested in and able to answer these questions vs the number of people asking them. That balance is crucial and perhaps hard to assess while these questions aren't a major fixture on the site.
One final thought is that in a few years, it may be that transcription software has progressed to the point that many music snippets can be substantially transcribed automatically. So it may be that if we hold off long enough, the demand will go away...