That's, uh, that's hyperdefensive as hug, actually. I actually noticed the term used was "mek'leth" in between when I rewatched it with subtitles on and (gasp) didn't even correct myself. Why do I get the feeling you've already read the comments about the class name....
So, yeah. The joke is in no way that they get the audience to be expecting something cool, so the show gets to disappoint them instead. I'd love for there to be some double reverse Uno card shot off the rim here, but it's just not in the show. The show just did a basic. It's as surface level as it looks.
It's La'an and Una who refer to "singing Klingons" as I recall, and neither laments "Klingon singing", but the concept of "singing Klingons". (Like, someone might resent showtunes as "white people music", but they're unlikely to find "singing white people" a novel concept. Unlike La'an and Una, Spock has by now experienced Klingon singing, and would not be laughing at the concept of "singing Klingons", despite probably hating it.) So I imagine we both agree that the characters don't know.
But I don't see any indication that the show expects the audience to know either. When we finally get the hail, the expectations of the characters that "singing Klingons" is a very silly thing are simply reinforced. As far as the crew's concerned, asked and answered. They aren't sitting there disappointed that the people they don't picture singing aren't cool when they do it. That reaction also certainly doesn't become any funnier to the audience simply because we know it's an unfair characterization. Quite the contrary in fact!
And think about the kind of Thanatos gambit you're proposing here. This is part of the grand finale number, and you think they want to disappoint the audience as a joke at this moment? It's complete nonsense. (That trick is used correctly by the way with "A Private Conversation," and where you expect it during the rising action.) Meanwhile, a year and a half after sea shanties were a TikTok trend is the least cool time to profit from a meme, but you know, you still very much do anyway....
No, they just missed the shot, it's okay, you don't have to convince yourself they're secretly winning four dimensional basketball. I'm glad Star Trek has participated in the tradition of musical episodes, but I'm only giving it the participation trophy it's earned for that. These are hardly the only problems, there's bad structure and missed gags everywhere, they're just the two things that are going to be nibbling the back of my brain for the rest of my life. Frankly, even between the two, the Klingon thing is just a missed (if obvious) opportunity, the Uhura thing is a crime against the structure of the musical.