Without a written constitution there is no limit on what a party with a majority in Parliament - which it can an achieve with a relatively small share of the vote - can do. Not a healthy situation.
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.
I think that there has been a degree of constructive ambiguity in the past. No more and it’s 100% clear now that the King / Queen would rubber stamp even the most extreme legislation. With FPTP, weaknesses in Lords appointments system, no written constitution etc then that’s a big weakness.
Not really. Since what you're calling for specifically is the Monarch to start blocking bills: it was established in the Bill of Rights Act 1689 (which would form part of the basis of a constitution if codified) that Parliament is sovereign, and the actual Sovereign can only act on the advice of parliament apart from when exercising royal prerogative, which is done on the advice of ministers.
One example of royal prerogative is proroguing parliament, which we've discussed. Another would be giving, or not, Royal Assent to a bill. The last time Royal Assent was refused was in 1708: https://en.wikipedia.org/wiki/Scottish_Militia_Bill and that was also on the advice of ministers, as they'd changed their mind about a law that had completed its passage through parliament.
In other words, what you're asking for has not happened in over 300 years.
The monarchy are supposed to leave the people alone to govern.