I don’t think that’s really a thing in the UK. In every election for over a century, when there are disagreements (and there have been many), the parties thrash it out and pick somebody to be prime minister, and that person drives to the palace for the Queen to make it official. It’s entirely ceremonial and I haven’t heard of any instance at all where she was actually involved in the decision.
There has been a lot of speculation over the years about whether Charles might be a more activist monarch, but I’ll be really surprised if he actually tries to exercise any of his theoretical powers. He might be a bit more outspoken in public, and do a lot more lobbying in private, at most.
Yes, there are question marks though that the Queen theoretically fulfills.
Otherwise, we might need a 'Supreme Parliamentary Council' to basically enact those duties, and if any members of Parliament didn't agree on the outcome, they'd take it to the Supreme Court who would rule on it kind of thing. Something that would only happen 'once in a century'.
Where there are Presidents, it's generally straight forward: the Dude with the most votes (of whatever type) is the Dude and that's it. There can be voting shenanigans but generally not outcome shenanigans.
I'm fine the way it is in the UK and Canada, I wouldn't change a thing.
If we want reforms, we can do that at more operational levels, aka 'governance by blockchain' to put it in 2019 Valley terms.
There has been a lot of speculation over the years about whether Charles might be a more activist monarch, but I’ll be really surprised if he actually tries to exercise any of his theoretical powers. He might be a bit more outspoken in public, and do a lot more lobbying in private, at most.