That and the election for the President is functionally independent of the House and Senate elections because they’re serving terms and controlling Congress doe

That and the election for the President is functionally independent of the House and Senate elections because they’re serving terms and controlling Congress does not mean controlling the Presidency and vice versa. In the UK, y’all put the leader of the majority party (or the leader of the senior partner in a coalition) in Parliament as your Head of Government which is essentially a selection process which is an internal party affair; and your Head of State is of course the King. I think for most Parliamentary systems (with or without a Monarchy), the Prime Minister or equivalent thereof is chosen similarly.