The monarchs are mostly a holdover from earlier eras, and are essentially powerless / symbolic in most Western countries. The important part is the way the legislative, executive and judicial powers are organised, and there are several ways to do that. Can you point to a republic that is obviously working better ? And is it working better because it doesn’t have a symbolic monarch, or because of that other stuff ?
We had a recent thread explaining parliamentary political systems. I suggest you have a look.