All the rich have to do avoid death , in this scenario, is share their stuff around a bit more. Doesn't sound that hard to me.
Biggest problem with death penalty, in my opinion, is the fallibility and corruptability of the justice system.

Its not that hard to see when people have accumulated excess wealth - at least in terms of physical stuff like land. I'd love to guillotine away all excess concentrations of power, but that's not as easy to observe, so start with land, and see if they get the message.