I'd suggest that having a funny face is an objectively bad way of determining whether someone is fit to run a country.
I only disagree that it's objective. As you said, plenty of people vote like this and are perfectly happy with their choices. You want for people like you and me to assert our superiority over their judgement. Annoyingly, this is only desirable to those already convinced.
I say it's objective because there are certain traits which one would fairly reasonably associate with being a good leader.
Having a funny face isn't one of them. Nor is someone's star sign. Or a load of other things which someone might use as a reason for voting for someone, or not voting for them. Unless someone can demonstrate that Geminis make better leaders then people shouldn't be using that as part of their reason for voting for someone.
Let me probe this a little bit. Let's say that we take the average "stupid" voter and educate them on what parties stand for. What happens when they start voting for completely idiotic policies? For the sake of a thought experiment, imagine that they're voting for something "objectively" terrible, and they're doing so in droves. A certain referendum comes to mind. Would you still consider this an improvement over funny faces?
Well. I don't think there are many objectively good or bad policies.
Without wishing to go all Godwin's Law, I don't think there's any party extreme enough to have "exterminate the Jews" as a policy, one which I think the vast majority of people would agree is objectively bad. Politics in this country is fairly middle of the road, the pros and cons of most policies can be debated. People want to pay less tax, but people want good infrastructure and a lot of people think nurses should get paid more. You can't have it both ways. So most policies have pros and cons. It would be nice to think that people voting have some handle on what the parties policies are, at the moment it's become little more than an X-Factor style popularity context. Style over substance isn't new of course, but I think in the era of social media and increasingly polished media soundbites it's got worse.