There's some sticky stuff in here, too.
People have vastly different sensitivity levels. 'That made me uncomfortable/sad/etc.' is doubtless true when someone says it, and it is absolutely a time for empathy, but I also don't think it's an entirely reasonable metric for 'do we take disciplinary action about it as staff'.
I'm being very specific for a reason: taking action and taking disciplinary action are not the same, and that can be especially relevant in in this circumstance. I don't consider, 'Bob doesn't like joking around the way you do with others, so don't joke around that way with Bob, OK?' to be disciplinary action, but it is taking action.
Being mindful of Bob's sensitivity is important, but if Joe isn't being a dick, or trying to harm Bob in any manner, Joe doesn't deserve a smack for it. He does need to know not to do that around Bob any more, and he needs to not do that to Bob any more. If he continues doing that to Bob, then he's being a dick, and deserves the smack upside the head.
That aside... using sensitivity alone is a concern to me. It's not something to disregard under any circumstances, but I've seen a number of instances of it taken to extremes I can't say I'm ethically comfortable with at all.
There are the obvious ones -- the person who can never lose or be intimidated or share the spotlight or look bad IC, for instance. They may be sensitive to these things as a player, and not want to experience them, but that doesn't erase them from being part of the game, and experiencing them part of fair play on the whole for everyone.
It's the less obvious ones that worry me more. 'Cry bullies' exist, and it's good to be mindful of them. I hate that term as I think it's gross, but it describes the concept well. Most of these folks do not have bad intentions, but are extremely sensitive to things in a way that does start to actively constrain others around them in ways that can cross over into uncool territory.
If Bob, above, can't hear anything negative directed at them under any circumstances, and complains any time they hear even the mildest criticism, there's eventually going to be a problem for which there is no easy solution. Bob feels what Bob feels, and respecting that on the human level is important. Making rules or taking disciplinary action based on what Bob feels in this case? Makes me very uneasy. It can't simply be dismissed as 'Bob is oversensitive, tell Bob to suck it up', but there's no real easy answer here that I can see, either, beyond telling people to simply not engage Bob in that manner, potentially over and over and over and over again as different people do this, and that doesn't seem like the appropriate answer, either.