Not sure there can ever be such a thing as an "automated website," as there will always need to be someone needed to respond to requests to confirm domain ownership as ICANN requests, plus over time things always change - new browsers require websites to be updated, yada yada.
I think of the predicament as the "Lone Wolf" syndrome - and I am (or was) in a very similar situation. I built my web site company all on my own and up until a year ago there was no one else helping run my 70+ websites for my clients. But I'm getting to the age where health issues arrise and I see former classmates dying and I began thinking of what would happen to my company if I died? It wouldn't be very responsible of me to leave my clients without anyone to manage their websites (afterall, that's what they paid me to do), and my husband certainly doesn't want anything to do with my company. He has his own (much larger) company, with 5 employees to deal with on a daily basis. And I have no children to pass on my business to.
So I began looking for an assistant - someone I could hire as part time help that I could train in all aspects of my business, from bookkeeping to coding. I actually found the perfect person and in one year she has now gained 90% of the knowledge required to continue my business without me. The other 10% will be learned in due time, as well. She's still part time, working 1 day in the office and the rest of the time on an "as needed" basis from home, which suits her (as well as me) perfectly as she is a single mother with a disabled child and our work situation permits her to put her child's needs first. A bonus for us both is she has a brother who is about to graduate with a degree in computer science who has also been tapped from time to time for side jobs and we know he's always willing to step in to help when needed.
Now I know my company will be taken in the event of my passing, as I have set it up that the company passes to her in the event of my death. My customers know they are being taken care of and that worrisome burden has been lifted from my shoulders.
This probably isn't a right solution for everyone. I'm just sharing how I solved my situation.