In a situation like that, it's a question of when, not if, someone will find you.
Also I just think you haven't thought through many of the problems you presented, most are easily solved.
24h guard? No, I'll make good use of zoneminder, moron detection, loud alarms, and a lot of cameras.
I mean I already have that and I'm not wealthy, for an American.
It's not hard to hide a place underground or in a remote wilderness when you have a lot of money.
Or just somewhere easily defensible, with one path up, that requires heavy equipment to alter.
As a billionaire, I believe I would have built some fortified positions to fire from without much chance of being hit myself.
Also I doubt I'll have any kind of obvious entrance to wherever I am.
Even on my own real-life property (I'm not a billionaire), I'm not just going to be like...in the living room. In an end-of-the-world scenario, I'll be camping discreetly somewhere on my own property, which is mostly a hardwood forest with a pretty dense understory. It wouldn't be any more inviting than hundreds of thousands of similar acres, so security through obscurity is pretty good.
I'll have access to all the tools I need, a house, my own water source. I'm not particularly likely to survive on my own permanently, but I think I've got a shot to wait out a storm for a year or two.