It seems unlikely that a dispatcher has any legal authority to accept whatever liability there might be. Still referring to the caller as a "nurse".
I don't know the law, but maybe the dispatcher doesn't need any authority. Perhaps the rule is that once the 911 call has been placed, 911 is liable for any instruction or action performed as a consequence. But then again, I'm no lawyer, and I'm just guessing.
I stopped on page 1, and haven't seen updates, but here's my take.
I worked as an EMT in CA for 7 years. 3 of those were with a company that did mainly transfers between hospitals, nursing facilities and associated health care places (PT, dialysis, etc).
First, if this lady was a nurse, in any capacity, then she's probably going to lose her license. She does have a legal duty to act, which means perform CPR, unless the patient had a DNR. Which in CA is very specific - a living will doesn't hold weight. There is a state-approved form that needs to be signed.
Second, I wonder if she'll be liable for a civil suit by the family, if she did have a duty to act.
Third, EMS "authority" varies by jurisdiction. Some places allow 911 to talk people through procedures, but this opens them to liability. So it's possible that the dispatcher was correct when they said that this lady wouldn't be liable.
And in CA, assisted living facilities typically are glorified apartments. You get help with cooking, sometimes medications (ie, dispensing), help arranging appointments and to/from doctor visits and what not. There's no actual CARE given regarding medical stuff. Skilled nursing facilities are the "nursing home" that most people visualize ... which in some parts of CA, is not much.