"The current situation only proves that the first past the post is now flawed"
On the contrary, FPTP systems are less flawed than any other system. You are making the error of thinking you are voting for a government, you are not. You are simply voting for your representative in parliament.
In is not a case of which group of people do you wish to support, it is a case of which individual do you want to represent you, to speak and vote on your behalf. And that is all you get to do. It is a mere curiosity that the representatives who pledge allegiance to one group can receive more votes overall than the representatives of another, that ends up forming a government.
And in any case, making the mistake of changing that part of the system would make no difference to the ability of the monarch to chose whomever they wish to lead their government.
What is not right is not the FPTP system, but the existence of parties in the first place. It is one thing for like minded people to get together to form agreements, as long as they continue to represent the views of their constituency. It is quite another to form an official party to apply pressure on those who join, to place the party's aims above those of the people who voted for them.