Why do the British still have royalty?

Best answer:

Because it works. The last thing Britain wants is some douchebag politician as head of state.

After 9/11 Bush's approval rating went up because the country rallied around the head of state. In the UK, the Queen fills that role and rallying around her is mostly harmless. The Prime Minister whose intelligence lapses caused such an event would not have such an easy ride.

It's separation of patriotism and politics and I cannot commend it to you enough.