This is a screenshot from a brand new run, actually following the exact order of events you suggested as a test!
I started a new game (not skipping the intro), got up on day 1, completed the bath task, and did the Tower Duties option at 9:00, which went directly from a black screen saying "You spend an hour fixing things in the tower..." to a black screen with the opening line of the game's intro, then on through the rest of the intro all the way through until waking up on day 2 of the game, having completely bypassed the rest of day 1. All I've done so far on day 2 was step out of the storage room so the text of the debug info could have a background it stood out against
(I also did another test following the same steps for a new game, where I did skip the intro, and the only difference in the debug info screen there is that Irith ARO was 5 instead of 6).
Additionally, it seems to me like it isn't actually game-breaking, just feature-breaking? The only thing it does seem to break for me is the tower duties feature itself; I first noticed the duplicate starting quest issue in a run of the game where I'd gotten far enough to have the portal room key, and seemed to still be able to keep going forward with the story, so the only thing the bug is doing to my playthrough is depriving me of any way of passing an hour if I have nothing to do but just want to advance time in the day, unless there's eventually going to be something that I'll need to specifically do the tower duties to unlock (or something that only happens at a certain time of day that I don't have enough actual things to do to advance time to get to it)?
Anyway, thanks for helping me look more into it, and if there's any more info you could use from my side of things, just let me know!