I'm unsure how the monarchy could be a check and a balance when the "constitutional settlement" is generally considered to include the monarch staying out of po

I'm unsure how the monarchy could be a check and a balance when the "constitutional settlement" is generally considered to include the monarch staying out of politics publicly - the monarch acts on the advice of ministers, which is to say the monarch is essentially a rubber stamp with several enormous palaces they can be stored in. There are checks and balances. When Boris Johnson unlawfully prorogued parliament, judicial review leading to a supreme court judgement eventually overturned the order, concluding his advice to the Queen was outside the powers of the Prime Minister. This is a fairly drastic outcome but Brexit was an interesting constitutional fuzzing campaign and we had a situation where the speaker of the house stretched the definition of impartiality quite a bit too. Other checks and balances include His Majesty's Loyal Opposition and assorted members of Parliament who may disagree with whatever bill is under consideration (the opposition are currently a bit useless, you may be forgiven for thinking they don't exist) and the House of Lords, who scrutinize bills passed from the commons (and do object to things quite frequently in spite of the irritation this causes to the PM of the day). It isn't perfect, but as the system is currently set up I'm not sure the monarchy can be regarded to have "failed" to check or balance the party in parliament. As to whether the constitution should be written down - I think it would probably be useful, but it is unlikely - and checks and balances still work as shown in the prorogation case. As for first-past-the-post and the UK's electoral system in general - again, yeah, a problem. I'm British and it doesn't matter who I vote for since whoever the conservative candidate is will be returned for my constituency, guaranteed. However, that's also not something the monarchy can change.