We can't really speculate what the US would be like as a constitutional monarchy, as it's just very culturally anti-monarchic. The Second Amendment and free speech, for example.
In a monarchy, laws often restrict people from insulting the monarch. Not in UK, I believe, but even British culture pays their respects to the king. As a result, the king's words hold a lot of power. A president can "talk down" to congress, but a PM is still a servant of the king.
Let's say someone like Sir Richard Branson decides to do a Trump. If he claimed that he had to do X for the security of the nation, the king would be able to call him out on it. As head of military, the king has access to all the confidential data. The Supreme Court and Congress may be missing data. As the PM has to get the king to rubber stamp military actions, the king still has the right to veto it.
In a monarchy, laws often restrict people from insulting the monarch. Not in UK, I believe, but even British culture pays their respects to the king. As a result, the king's words hold a lot of power. A president can "talk down" to congress, but a PM is still a servant of the king.
Let's say someone like Sir Richard Branson decides to do a Trump. If he claimed that he had to do X for the security of the nation, the king would be able to call him out on it. As head of military, the king has access to all the confidential data. The Supreme Court and Congress may be missing data. As the PM has to get the king to rubber stamp military actions, the king still has the right to veto it.