How does the doctor and nurses know my aunt (97) is returning to a safe situation?

If she was able to call you on the phone that is great, she can still use phone. If she were my aunt, I would be calling APS first thing Monday, and in the meantime, does your aunt have a church membership, even if she hasn't gone in awhile, or is there are church you yourself are a member, where you can call upon their Hospitality committee, or somebody, to stop by with a bit of food? I only mention the churches because they are able to mobilize someone within hours, whereas APS is going to take awhile (and might just remove her, on their schedule not hers). Or are there neighbors you could call who could do same? Long-term it does sound like she will need significant help to stay in her home, or if a move to NH is best (who is her POA?) She will still need significant help from several people to facilitate the move. Best of luck, you are busy enough with your own parents, I hope there is other family to step in now.

So what happened to the family member who was supposed to be keeping an eye on your aunt? As a first step, I'd give that person a call and find out what's going on - without panicking, by the way. There may be very good explanations for what's happened that will become clear; but you've had a shock, and you're stressed anyway, so I'm not blaming you for being very worried about her.

Kudos to your aunt for getting away with no broken bones! I thought I was going base over apex this morning on ice, it can happen to anyone - she's done very well to stay in one piece.

If she is alert, knows her name and where she is and the date, they have no basis to keep her against her will. Look up "mini mental exam".
A classic dementia patient would get lost before she got two houses away.

She is 97 and had fallen . Does this not cause her to be considered a risk? They are professionals' can they not notice dementia? My aunt did not have children, and the other family members do not want to be involved . I am so overwhelmed with my role as a caregiver to my parents(my mom is a stroke victim) and I am a mom of three young boys, I am also a wife. She is 97 not too surprising she has dementia.

The ER cannot stop her if she knows the way home.

She has dementia. She may have told them that it would be ok to send her home in a taxi because in her mind it was. Sounds like she needs to see a neurologist for a full cognitive evaluation.