How about, instead of asking "Tell me about a time you...", and you presuming to understand the situation so well that you can make judgments about nuance based on the off-the-cuff example you asked for, and trying to cut them short from "rambling"...
You instead ask them "Say you have a situation like X; how would you approach it?" And you can change the situation by adding information, "What if the report responds Y?"
Then both people are operating from closer to similar information about the situation. (Though it's still possible that the interviewee understands something about these situations in general that the interviewer doesn't.)
This also avoids dredging up past unpleasant situations (someone with more experience will have handled more unpleasant situations, but that doesn't mean it doesn't invoke a somber mood, if they're not acting or oblivious).
It also means they don't have to also think about how much they can say under NDA and being discreet about personnel matters (while a poor interviewer might take hesitation or choosing words carefully as interviewee trying to put themselves in the best light or keep a fabricated story straight).
An expected objection to this approach of spinning a hypothetical situation is that candidates might just say what they think are the correct answers. But knowing the correct answers is at least half the problem. And what do you think many candidates are doing in the interview anyway, if they are the kind to know the correct answers, but not follow them in real life.