This is also reminding me of a woman my DH worked with or 2-3 years in the early 2010s. She was about 12-13 years older than us, (we were around mid 40s, and she was mid-late 50s,) and she seemed to be in poor health a lot. She had been a heavy smoker and had all kinds of smoking related illnesses. She had one son too (like my great aunt,) but this woman, and her son, hadn't spoken for 15 years as she hated his wife.
She left the workplace around 2014, and DH stayed in touch with her, and we would meet her every 6-8 weeks or so for a coffee. (She lived 20 miles away.) I found her quite annoying. All she did was whinge and moan, and especially about her son and his wife. I didn't want to go with DH, but he begged me to.
Long story short, after about a year, she started to do what the OP is doing. Edging towards us being POA on her death, and wondered if we would sort her funeral when the time comes, and would be her carer when she became unable to do things herself. DH had already run her to hospital appointments 3 or 4 times which pissed me off as she was capable of driving. He asked me to take her a few times, and I said no. I have my own family and job and DC and commitments, (and so did DH!) and it annoyed me.
She had a house of her own - worth some £60K at the time, (a tiny cottage in quite a rough area,) but she had £40K or more of debt. (That we knew about! I suspect she had more. She never stopped spending!) She said 'ooh, you'll get a little pay off in my will if you look after me.'
tl;dr, after I said to DH (multiple times) 'No fucking WAY are we having any part of this' and DH had to admit defeat, he phoned her and said 'no sorry Pauline, we can't do it. It's too much for us to take on.' She got really snarky and was proper pissed off. And like my great aunt, she just cut us off!
I was delighted. DH was really disappointed and hurt. I said 'what did you expect? The woman has cut off her son because she doesn't like his wife, she is not a nice person! She has been using you, she was attempting to use me, and when it all stopped, she's thrown her toys out of her pram.'
Don't know what became of her. Don't care. DH never contacted her again.
@MarilynAE I am not saying that YOU are anything like this woman. It's just bringing these things back (from the past) that have happened, in a similar situation.