I doubt the very rich are not aware of this, so would have come up with ways to ensure continued loyalty for their own safety. They may be immoral, ruthless, sadistic, etc, but doubt they are stupid.
For all we know they may have explosives surgically implanted in their servants or some such kill switch they can control.
Their biggest danger is survivors blocking them in. So they will have secret exits etc as part of planning.
I disagree. The very rich can be stupid—because of entitlement and vanity. They also become weak and helpless from the luxury. And they can’t handle being alone. I don’t think they’ll last an hour after TSHTF.