The reason is because of the Manikins who were only released in the 12th Cycle
Exdeath releases the manikins as foot soldiers for Chaos. Due to their nature as mindless automatons, Garland remarks they will continue to fight even when their opponent is defeated and weak. Thus, if one is overwhelmed by manikins, they can be pushed to the brink of death and risk not having the strength to be revived in the next cycle.
Source - Manikin
Kain informs the Warrior of Light of what he learned and the two reach the agreement they should defeat the other warriors of Cosmos before the manikins do, who are the only ones that can permanently kill the warriors summoned by Cosmos and Chaos.
Source - Kain Highwind/Dissidia
When a Warrior of Cosmos or Chaos is defeated by another warrior, they will be revived in the next Cycle minus the memories of the previous cycle and their own personal memories from before the war. however should a Manikin defeat a warrior there is the very high risk the warrior is permanently dead which was the plan of the Warriors of Chaos. the plan of Kain and the Warrior of Light was to control the potential damage and losses in preparation for the next cycle.
the Cosmos Crystal's purpose was only to allow the Warriors of Cosmos to exist should she be destroyed, and with her proper destruction being a sign that it was the final cycle she had the hope that the Warriors would then face Chaos and be free to return to their worlds.
The Emperor only created the Dark Crystals for his own ambition to become a god after Chaos's destruction which one can assume that the Dark Crystals worked in the same way as Cosmos Crystals by allowing the Warriors of Chaos to survive after Chaos's Destruction (Though the Emperor is defeated beforehand)
NOTE: i say Dark Crystals over Chaos Crystals as the Cosmos Crystals were fragments of Cosmos's power. the Dark Crystals are not fragments of Chaos's power as they were based off the Cosmos Crystals Jecht would have received (being a former Warrior of Cosmos)