> If the executive doesn’t have parliament, they lose the executive.

Not automatically. A minority government of course more at risk of losing the confidence of parliament but it's also a powerful incentive for such a government that want to survive to use cooperation and compromise with the opposition.

In a British system, isn't the head of state hereditary, and in theory has no majority or minority because they have a divine right to be there?

Which they are apparently are, given how many chancellors the UK went through. Thinking for yourself is a lot easier if the guy you're thinking about is removed from power when you've finished your thoughts.