We are straying a bit from the original question, jake, but I do find the question you pose is a bit difficult.
We have a Head of State (the Monarch). I am not a republican and would like to retain the Monarchy, but apart from ceremonial matters, and theoretical powers of veto over government activities, HMQ plays little part in “running the country”.
There is not only the matter of the Prime Minister (or President, call it what you will) to consider, but also of the remainder of government. The easy way would be for Ministers (including the PM) to be selected by all the MPs. But of course the party with the majority of MPs will certainly see all the posts filled from their numbers, so it is a non-starter.
I think the issue is less about the selection of a leader and Ministers, but more about the influence the Whips have over MPs voting habits. They must be allowed more freedom to vote according to their constituents’ wishes, whether or not this complies with the party line. At present voters have essentially two packages of policies from which to to choose. But different bits of each package suit some and not others and voters should be able to rely on their MPs to vote according to their needs on each individual issue without being whipped into line.
At the moment they cannot, and, short of proscribing all political parties, I don’t see how they can.