It does, actually. What matters is that there is a king, not who is the king.

The saying works because, as you say, it suggests a successor, but the successor has to use the same title, because what people want is a new king, so that nothing changes and they can live as they did before the king was dead, not a revolution with a civil war tainted in blood.

If you do volontarily change the title, it's because you think the new one will be better, which is pretentious.