"As in my original post, when it really mattered - the prorogation of Parliament - the Monarch proved to be useless as a defence. We had to rely on the Courts, not our Head of State, to find the prorogation to be unlawful."
'No, you're simply misunderstanding what the Constitutional role of the Monarch is - to deny power to any would-be usurper and to be a figurehead for the UK under the Government's orders.
'It is the role of the Supreme Court to rule on constitutional matters. Even the Scottish Parliament was found to have acted beyond their powers.
'We have a Constitutional Monarchy and the Monarch has a passive role except for the highly unlikely situation aforementioned.'
I am not misunderstanding anything, I am saying that the current model does not and cannot work both in practice based on experience, and on its own terms. How would the Monarch deny power to a usurper or dictator in practice? How would/could they do that? Who decides whether an individual is a usurper and at what point?
"Returning to comparisons with other countries; many have written constitutions and elected Heads of State with real power to hold politicians accountable to the constitutions and act as effective arbiters of the democratic system."
'You're suggesting here that the UK doesn't have a written constitution which is completely wrong. We actually do - it's written in Acts of Parliament. Unlike the United States, our written constitution is uncodified which allows the UK to change constitutional laws very quickly if needed.'
There is no actual "British Constitution" just a set of rules and regulations constituted by jurisprudence and laws (English and Scottish law), and by various treaties and international agreements to which the UK has signed up.
The lack of a written constitution caused a major constitutional argument about what a government can do without the consent of Parliament. Just to summarise here, the problems this led to following the 2016 Brexit referendum:
Constitutional experts agreed that Parliament had to be consulted, and vote, before the Prime Minister could activate Article 50 of the Lisbon Treaty, to take the UK out of the European Union. However Theresa May (PM at the time) did not want a debate, or a vote, in Parliament before Britain left the EU, and sought to go ahead with Article 50 regardless.
In July 2016, a group of citizens appealed to the High Court to stop the government activating Article 50 without Parliament's consent.
The verdict was confirmed by the Supreme Court on 24th January 2017; Parliament must vote before Article 50 of the Lisbon Treaty could be triggered. It did so.
Nevertheless, Parliament was not bound by the result of the Brexit referendum, which was consultative, not legislative. Before the referendum, three quarters of elected MPs were against Brexit, and until the 2019 General Election, it remained possible that Brexit would not actually take place; Parliament had the power to veto it.
Vetoing Brexit, however, would have been a dangerous step. If Parliament had vetoed a process that was approved (however marginally and on the basis of however many lies) by a popular referendum, it would have sparked a massive constitutional crisis and possibly serious public unrest. In the end, Parliament held up the Brexit process, but could not stop it. While some Conservative MPs rebelled, enough remained loyal to their party to ensure a situation of stalemate (impasse) in Parliament, where MPs could not agree on a way forward.
Eventually there was a new General Election in the autumn of 2019, and Boris Johnson secured a majority on his promise to "get Brexit done". Although the Conservatives secured the highest total number of votes, overall fewer voted for them than for other parties. In some countries that might have resulted in (the equivalent of) a hung parliament, however with our first past the post system that was enough to secure a huge majority for the Conservatives. With this Parliament approved Brexit, and the UK duly left the EU.
The debates and arguments did not stop there. Later in 2020, Johnson proposed a law that would allow the UK to renege on parts of the international agreement signed with the EU concerning Britain's terms of exit from the Union (the Withdrawal Agreement). So apparently, despite the Constitution being constituted in (among other things) international treaties, those treaties can be broken at will by the PM of the day and there is no mechanism for stopping this. Parliament is accountable, but not the Government.
No wonder there is lasting anger from many people about Brexit. It has opened up the deep divisions among people within the UK; how we are meant to repair that, and how the Monarch is meant to act as a 'unifying' figurehead I don't know. BTW Johnson's actions were severely criticized by ALL of Britain's living former Prime Ministers (3 Conservatives, 2 Labour), as well as another former leader of the Conservative Party.
There are therefore strong arguments for the need for a UK constitution which are made by many. The attached article is helpful for further explaining the issues: www.ft.com/content/bf81ea49-96b0-4329-bbaf-f58fdf6e0630
You cited a number of existing models for the role of Head of State. I agree that there are pros and cons with each of them; I point out that in other countries the HofS would have the power to force the Government of the day to comply with its own laws.
I would disagree with you that the UK's checks and balances work well: imo they do not. Or nowhere near well enough.