Why the obsession with this at all?
What's wrong with just government with a Prime Minister? Presuming that the monarchy had been abolished?
The prime minister is not selected by the people, but indirectly by MPs, who can change their minds between elections. Nobody voted directly or indirectly for Sunak, Truss, Boris (before his re-election) or Brown (who was booted out at the first electoral oppportunity).
We need to do what modern countries do and have a head of state (president) who is directly elected, who runs the country. The prime minister is then merely the head of MPs, who reports to the president.
Direct election of the president would allow outsiders a free run at the top job, it could be a good way of breaking the two-party domination of politics that is currently leaving many voters picking the least bad of the two very similar variations of the same policies, both parties are currently two cheeks of the same arse.
General elections are currently corrupted by our system. Many voters don't even know who their local MP is, but vote for them personally because they're associated with the voter's choice of PM. This is wrong, we should vote for our representative in parliament based on how good they will be at taking care of the issues that concern us locally. This is why we have so many MPs who don't take any more than a token interest in local issues and support the party line even if it's against the interests of the people who voted for them.
In several cases in the past, MPs have been elected on the strength of the party they represent. Then they switch parties after being elected. This is allowed by the current system, as most voters think they're voting for a party but in fact they're voting for a person who is currently a member of this party (but can do what they like after election).
When there are world leaders summits, our PM goes to represent us - not our head of state. He is one of the few at the summit who has not been elected by the people, and probably the only one who reports up to someone else above him. Those from other countries have the mandate to legitimately represent their population, as they were given their job by the people, and they are ultimately the person at the top.
We currently have an unelected head of state that nobody has voted for, and a PM that only a few MPs have voted for. Neither is directly accountable to the people of the country.
The presidential election should be for our choice of national leader, general elections should be for our local representative in parliament. The PM takes the concerns of MPs to the president.