In general, as armour got heavier and more effective, slashing swords went out of fashion as it was too difficult to cut through armour. Polearms with greater leverage, "smashing" weapons like hammers and stabbing swords like rapiers evolved to negate the protective attributes of armour.
The problem with a "dead blow" weapon is the force is distributed in both time and space, and diffused over a wide enough area that you will not be able to deliver a blow that would take out the opponent. The best you could hope for is to knock them down (and maybe follow up with a dagger), or if you are lucky, a blow to the head might stun them long enough to subdue them, or give them a concussion and put them out of the fight. Of course a knight or man at arms is wearing a great helm, balanced on a padded ring (a primitive suspension system, much like modern helmets have padding or straps to keep the helmet proper away from the head), a layer of chain mail (the coif) and possibly a leather skull cap as well, so it is easy to see why war hammers or halberds were favoured. Even a mace was usually made from multiple triangular "blades" around a central shaft with the points out to concentrate the force of the blow.
The best use of such a weapon as a dead blow flail would be if capturing the lower ranked levies is somehow important. A spiked flail such as the one pictured would cause lethal damage to the peasant levy called up into battle (generally unarmoured and trying to fight you with a pitchfork or billhook), so a dead blow flail would knock them flat with maybe broken bones or concussions, allowing you to scoop them up as captives. A team of people would be involved, one armoured person to wade into the mob and start knocking them down with the dead blow flail, while the rest of the team rushed in and grabbed the captives.