As a Brit, I like the UK system over ones with political head of state because it is untested in democratic times.
That is, the King can sack Parliament, and at the same time, Parliament can sack the King. But the King can't sack Parliament and take over.
So it is potentially a system in tension in the event of any dangerous scenario. If either side oversteps the mark, there is a panic button that can be pressed.
And this is untested because we do not have a written constitution. Although the late Queen did sack the Australian government once (without knowing apparently), and a few others.
It's untested because we do not actually know the limits of power, because , again, no written constitution. The Brit Armed forces, police, court, civil service etc, pledge allegiance to the King, but it is Parliament who run them, and more importantly, fund them. But at the same time, all is done in the name of the King.
And I think if this was tested to the max, it would result in a total collapse,. It is just too dangerous for either side to push too far.
It is, to me anyway, a pretty solid ultimate check and balance.
Both sides only have legitimacy because of the other.