Yes, I can see that the complexities of the separate election of the Prime Minister would probably make such a procedure unworkable. However, some democracies get round this problem by electing their head of state directly as well as their parliamentary representatives, and that seems to work. Others rely upon a system of elected federal government.
The problem with the UK is the lack of a proper, modern, constitution and reliance instead on a mystical hereditary monarchy together with a fudge of common law, acts of parliament and tradition. The role of the British PM is very loosely defined and leaves incumbents with powers to do all sorts of things - good and bad, almost like a President appointed only by his or her own cronies. That has got to stop.