On the one hand, I disagree with the premise that some languages are "struggling to modernize" in general. The most popular music in the world is somewhat generic "pop" music, found really everywhere, and that's quite "modern". There are also other very recent (in that sense of "modern") styles like death metal that actually are especially popular in some non-English languages (German, Swedish, etc.), even among English speakers.
On the other hand, the globalization of English, along with the extreme popularity of English music, now means that English music is overtaking other local music everywhere, even in countries where there is no (immediate) linguistic threat of English taking over the local language (e.g., France or Spain). The situation is complicated and mixed, so at the same time you also do find some music in the local language now copying English trends (like pop, and other styles) rather than continuing traditional music from the local culture.
Beyond the linguistic and social factors, there's also a huge branding and marketing perspective: it might be more appropriate to compare the spread of English music to the spread of Coca-Cola than to the spread of the English language in general. What do radio stations play? It seems that everywhere in the world you can turn on the radio and find English music. This creates a feedback loop: English becomes normalized as a language of music. Without the big labels and money behind local bands, they may struggle to get airtime or gain popularity, although of course the opposite factor of local interest helps them too.
Anyway, as for purely linguistic issues, there are some factors, but probably not as broadly as you suggest. The relevant literature would probably be about how poetry varies by language, and local traditions conforming to linguistic features. For example, rhyming is a feature of English poetry (and less often but still frequently in English music), but in a language like Latin where the endings of words are so often the same there may be other strategies like meter. These of course have an effect on local music. My guess would be that any complex traditional music styles might have features particularly suitable for the local language, and then they might not translate well into English. This is of course why pop (and perhaps death metal) translate from English so well: there really aren't any (linguistic) rules, so any language can fit. But that's not true of some local traditions. Related to this is the issue of popularity versus tradition, where any particularly limited and traditional musical style may fall out of fashion as younger generations look to new trends-- a somewhat natural cycle, but interrupted and shifted when those new trends are international.
In the end, music is a product, and it will be whatever sells. That doesn't mean that the current situation is the only one that would create sales, but it is hard to interrupt. So I'm guessing that's part of the issue: the current model is basically selling English music around the world, and that works (in some sense, at least in terms of profit), so it's hard to replace that with local music instead, although that would also likely work.
This reminds me of a German filmmaker visiting my university to present a film she had made. It was in German, with subtitles for us. But all of the music was in English. Afterwards during the Q&A, I asked about this choice: why choose English music? There are various answers I can imagine, such as capturing this zeitgeist we're talking about here. But her answer was surprising, and perhaps insightful: she said she didn't even really think about that, she just picked the music (which happened to be in English). She even seemed a little surprised by the question, as if it was obvious.
