Yeah interesting question. I actually don't know if it makes much difference, since in countries that ARE constitutional monarchies as you describe, the head of state (king, queen, governor general) is so invisible as to be almost nonexistent. Which is to say, the average person probably thinks the prime minister IS the head of state, if they think about it at all. And the prime minister is most definitely partisan. A very public, active and high-profile head of state can be a non-partisan force for good, however. Trouble is the head of state is often a useless cipher.
I do however think that the effective head of government should be tied as closely as possible to the body that elects him or her - ie. should sit in the elected chamber as a member of that chamber. You need as many brakes on executive power as possible. I'm not really a big fan of 'efficiency' when it comes to the important democratic questions. Governments in power tend to launch ambit claims, not good if they come to power with a zealous ideological agenda. You really rely on the machinery of parliament and an upper house to kinda slow down that sort of rubbish. That's off-topic to the main question of course, but well, I don't care.