Firstly, my current rough understanding of what 'analysis' questions are allowed is - we don't allow anything where having 'the score' of the piece would answer the question. To be fair, I think the users of this site are fairly consistent in applying that idea - and hence the question you mention falls foul of that, because if you had 'the score', you'd be able to see what the key was, and the notes in each chord.
I put 'the score' in quotes there because, for many pieces of music, there isn't a good-quality score representation available. When I talked about this in SE chat before, IIRC we discovered that there wasn't a score readily available for about half of a random selection of popular works - so the logic of not allowing 'transcriptions' can't reasonably be based on the idea that a person could just get the score - as often that's not actually possible.
One reason that's sometimes given for not allowing transcription-type questions is that they wouldn't be useful to future readers. I don't think this argument necessarily holds up; wanting to know how to play an existing song is a common need amongst musicians; I imagine there are a lot more musicians who will want to know the chords to Kylie Minogue's Golden than want answers to some of the other questions asked on this site. At best, you might say that the usefulness will come down to the popularity of the song, but even then, people might be interested to come across new songs they haven't heard before.
Another argument for allowing transcriptions is that stack exchange users are encouraged to
Focus on questions about an actual problem you have faced. Include details about what you have tried and exactly what you are trying to do.
And "I am trying to figure this out and I can't" is very much along those lines. Sometimes on Music.SE we seem a bit leery of helping people with their real-world, individual problems; we seem to want to pay more attention to the very general purpose, 'library of answers' type question. And there is value in that, but there's also value in the specific. Sites like interpersonal skills and language sites are full of very personal questions that for whatever reason, many other users find value in. Sci-fi is full of story-identification questions that you might think would be useless to others, but they're often very well-received and appear in Hot Network Questions.
One area where I particularly agree with you is:
To me, it seems arbitrary and, as we know, chord progressions are a fundamental part of popular music.
Absolutely. "I have this audio and I want to work out what notes to play" is just as much of a real and present problem for people doing music as "I have this score and I want to do a roman numeral analysis".
There's also the argument that the 'teach a man to fish' approach is more useful. In many cases, I agree - working out a key, for example, often isn't too tough, and we do already have some decent answers giving some good ideas how to do that. However, the "teach a man to fish" idea only has value if you actually do it, and I haven't actually seen any really good general answers on this site on how to work out chords. There are a few good bits of advice here and there but not much that I couldn't tell you, and I'm far from brilliant at working out chord progressions quickly.
There's also the argument that we're not trying to replicate existing archives on the internet (or elsewhere) of musical transcriptions. Again, this is true - but that doesn't necessarily prevent us from stepping in where there is a problem with what's already available. I'd see that as analogous to a problem that SE language sites have - they want people to be able to ask about words, but not take the place of dictionaries. And I'm pretty sure they have ways to solve that.
One possible argument for not providing transcriptions or parts thereof might be if there would be likely to be any legal issues. But the issues there would be the same as with reproducing transcriptions in questions for the purposes of more 'substantial' analysis, and that's not something I've heard any concern about.
That's probably enough rambling - Just some thoughts, and I'm not expecting to change many people's minds, so I won't go into any thoughts on how we could service these questions without devaluing the other content on the site. I'd like to think we could do it, if we put our heads together though!