I guess I should have clarified a little.
If a weapon is too slow, then don’t use it. This varies with your skill and strength, so for example, that axe, for you, sucks ass, but if you were stronger and more skilled, it would be good.
This is because if you are too slow, zombies have an easier time catching up to you and cornering you when in groups, and even if you kill with every swing, those groups will be able to get several hits in while you do. It also makes fumbles and misses very painful.
Notice the emphasis on too. Once a weapon is under ‘acceptable’ swing speed, even if it’s a little slow, I start looking at other qualities. That Zweihander is not something I’d wield at low skill/strength, but once skilled enough, I’d probably grab it unless I’m primarily a ranged character (too big to be wielded fast for a backup weapon), or I’m running with eskrima or niten.
Another similar caveat is if it’s too weak damage-wise, like for example most knives, I’d probably pass it even if they play nice with points 1 & 2 and I’m looking for just a backup weapon. Even high swing speed can’t make up for not being able to kill the enemies fast enough, not being able to punch through armor is also a concern, unless you’re carrying a backup (or primary) weapon to do just that.