When in chat, time is displayed in a localized format. This is nice - I know what time it is where I am. When looking at transcripts, though, the site displays them in system time. This is not very nice - math is hard. Seemingly every other timestamp on the SE network is displayed localized, I would love it if transcripts were too.
I understand that it displays a "quick jump" list to jump to any part of that day's transcript (in system time). It may be a bit tricky to implement that in a clean way, but even just localizing that and having it "wrap around" midnight I think would be more friendly than not localizing transcripts at all.