Since the chat system has received a bit more love recently (thank you @balpha), it would appear now is the time to make any suggestions regarding additional functionality.
Chat is still an important part of the system (albeit it secondary) and has an emphasis that's it public, so should be available for all - but is has no publicly available "real" archive as such.
I have looked at:
However, they appear to have received little attention from the dev. team. While technically it's quite easy to crawl the transcripts of large rooms, there's still a lot of network traffic to/from the SE network to do so and too-oft repeated requests may well trigger some throttling/ignore or otherwise warning.
Instead of using websockets or scraping transcript pages, could it be possible to make a room’s transcript available as part (or separately but similar to) of the SE data dump process?