Ok, so I really suspect the poll numbers are skewed by people from the US, where a cop without a gun is considered ridiculous.
As far as that video goes, firstly if they'd had guns, then what? I'm not convinced that shooting a mental patient for being mental is entirely justifiable, though granted, yes, he was posing a public danger, which is why he was surrounded by police. I would like to hear a solution to that situation which does not boil down to 'he's a loony, so shoot him'. Secondly, he had a machete because that was the most dangerous weapon he needed to outmatch the police. I suggest that if he knew the police came armed with pistols, he would be likewise armed, perhaps with something more randomly lethal, a shotgun or machine pistol. Rather than swinging a blade around wildly, he'd be firing off bullets or shot. How is that an improvement? And yes, supposition, but it's an arguable point whether US cops are armed to match the criminals, or vice versa.
Accuracy is really not an issue either. I'd like to be sure anyone issued a firearm can shoot straight, but it's far more relevant that they can make good decisions about who needs a bullet in them. Unarmed, police are required to solve problems in far less lethal ways. This is a good thing. I don't want to accuse anyone particularly of gun fetishism, but it does make me wonder how badly the argument is affected by the assumption that shooting bad guys is a good thing.
That said, I am a fan of options, and given the distinct possibility of armed opposition in some rare circumstances it seems prudent to give a police force limited access to weaponry, on the condition that it can be proved necessary in each specific case.
I do not trust anyone to effectively deploy armed response against civilians. It gives far too much scope for needless violence, and 'last resort' options tend to slide inexorably up to 'first response'. It simply will not end well.