I think a key difference is that in parliamentary republics, we usually now how the candidate (that the parliement votes on) has been chosen: it's usually by wi

I think a key difference is that in parliamentary republics, we usually now how the candidate (that the parliement votes on) has been chosen: it's usually by winning the elections as the head of the party with the most parliement seats. In this case, the process is completely opaque. I would say that it would be pretty weird if people say, here in Canada, voted for their MPs (good!) but then said MPs could only choose between an arbitrary list of candidates to be Prime Minister, chosen in a private meeting with 0 explanations.