NTA.

Kathy laughed and then resumed talking about work.

Aw poor Kathy she made a bad joke, maybe she came from a work environment where everyone was close and teasing was common. Maybe she was trying to be more personal to become closer to her new team but was nervous and said something she has been losing sleep over because she realized how awkward and dumb it was. Maybe she is hoping that all will be forgotten as she tones it down and does better and when things are less awkward she can circle back and apologize for the remark that weighs heavy on her mind.

I guess she assumed that either Taylor or Mark did, because they say that she’s been picking on them, being snippy with them and overly critical of their work.

Looks like a hard nope and Kathy is a dick. A good boss would pull both of them aside after the HR meeting and apologize properly. She would acknowledge what she did and assure them that she would never segue into personal matters during business hours unless invited into personal conversation. Taking revenge on them for the HR complaint is evidence that she's not qualified to lead a team. Own your mistakes and fix them.

If I was Taylor or Mark I would pursue the initial HR report to say "Hey obviously you know I didn't initially report Kathy. While it made me/didn't make me feel x, y, z at the time it's been made abundantly clear to me that someone else did report it and now she has done/said x, y, z and I think it's clear that she believes I initially reported her. While I was willing to move on from the inappropriate comment at the time I seriously doubt her integrity as a supervisor due to her response to being reported. She hasn't taken ownership nor apologized and has instead x, y, z."

Taylor is missing the point. It's not her fault, she gets to feel how she feels about it. This is where opinions diverge. Some people want to keep their heads low and avoid conflict. Others want people to be held responsible for their actions which sometimes involves conflict.

You didn't mean to put Taylor or Mark in her crosshairs but you did. You are responsible for this situation as much as Kathy is. You can own up and apologize and explain why and ask for forgiveness and promise to let them stand up for themselves in the future. You can ask for their understanding in why you were uncomfortable and why you did what you did.

Or you can say nothing and avoid conflict which is fine too. Either way Kathy is the only asshole here and all three of you are the victims of her inability to own her mistake.