You've had a shock and it's normal to go into lots of 'what ifs' in the circumstances. But you've really nothing to feel bad about. She was a capable adult, who made her own choices and lived her own life. Part of that is that she died her own death.
In the nicest possible way, how did you think she would die?
Peacefully at home is a pretty good way to go, for anyone. For someone who dislikes close human contact, being fussed around in hospital by staff and well-meaning relatives, would have been pretty dreadful.
Being given some warning, via a hospital admission, might have given you the chance to fuss around, bring her stuff and feel useful. But frankly, that's all about you and your feelings, about circumstances giving you the opportunity to feel good about yourself; nothing to do with her and her feelings. What do you actually care about here? How things were for her? Or what other people think of you?
Is there some concern that her death was not peaceful and could have been prevented or made more comfortable if medical attention had been sought? She was clearly a competent, independent person, perfectly capable of making a doctor's appointment or calling an ambulance. Is it possible that she didn't realise how ill she was and that someone else would have prompted her to seek medical help? That would be the thing to think about with other relatives.
But honestly, you can provide panic buttons or emergency phone numbers but people are probably no more likely to use those than to call an ambulance direct. Unless you start calling people weekly for a chat, or dropping in on them, or making sure someone else is doing this, you will never notice the sort of early stages of illness, or rapid decline, that would prompt action.
So the question is, do your other relatives want that sort of close, checking up and chatting regularly relationship? With you? Or with someone else? Or are they actually happier living their independent lives and taking the risk of becoming incapacitated alone? That's a choice for them to make but you can certainly start that conversation.
Btw, practical point but the normal thing is that the deceased's estate pays for the funeral. Why are you paying? Who is her executor?