In this case, HR wasn't the problem. They were helping and tried to improve the situation. The problem is that there's only so much HR can do when the top executive chose to completely disregard them.

I'm biased, my wife works in HR but I've heard multiple stories from her where she helped solve problems by acting as an intermediate and deescalating the situation.

One of the major impetus of HR is to comply with laws regarding discrimination and ensure that the company doesn't engage in behaviours that would result in them being either liable or having a PR problem. This means solving those kind of issues and that sometimes involve batting for the employee with the executive team because they know that it's in the best interest of the company.

In this specific case, OP is a manager and passing up the chains issues that have been signaled to her. In a well functioning organization, this is absolutely the correct response and it's part and parcel of a manager's job. Involving HR early with a clear solution to deescalate and improve the situation (as described by the first case from OP) is great because this is what's best for the company. If the employee had transferred to the new team there would have been no basis for a lawsuit.

If you have a good relationship with an HR person, and trust them enough to ask for informal advice, some positive action can follow. If you want to go the formal route, if you use words and phrases that sound like a lawsuit in the making, expect an action to your disadvantage. HR is there to protect the company, not you. An obviously disgruntled employee crying foul is immediately perceived as a threat to the company, regardless of whether they're objectively right or not.