Nanarosie, whether your aunt has money or not Social Services have a legal obligation to make an assessment of her needs, even if she then has to pay for any care that results. Social Services will use the 'self-financing' card to avoid fulfilling their statutory duty whenever they can. Dont let them. Sadly my experience of SS is the same as HildaW's. There is no situation involving the care of the elderly that Social Services cannot make worse.
However, even allowing for the very natural grief of your aunt at the loss of her husband, refusing to pay for any care that she needs is ridiculous and expecting a niece that she hasnt seen or shown any interest in for 30 years to provide all the care she needs for free is outrageous. Stick to the phone calls and go no further. You have never suggested that your aunt has any mental problems and from what you have said about her and her husband's lack of friends and family contact it would suggest that her current behaviour is the way she and her late husband always behaved, which is why she is in this friendless position. If she chooses to continue to act that way she must accept the consequences.
As for pushing her off a balcony. I have looked after four very disparate groups of relatives at different times over the last 15 years and when I felt like that, and believe me I did. I relaxed, indulged the thoughts, and then let them flow out of me. It was calming and cathartic and enabled me to keep going.
Nanarosie, Your health and wellbeing and that of your nearest and dearest come first, an aunt who hasnt seen you for over 30 years comes way down the list.