If the CEO says they'll resign, people will say they're taking the easy way out, and anyway, it's no skin off their back because golden parachute (if they were a good negotiator when they were hired) or anyway their salary was probably high, etc.
It's an unwinnable game in the court of public opinion, so why do it? Because it signals to the underlings that the CEO is not expecting resignations from them.
It's an unwinnable game in the court of public opinion, so why do it? Because it signals to the underlings that the CEO is not expecting resignations from them.