Here's a simple solution: When a team member goes down, it reduces the max health of everyone else by 25%, stacking up to -75% if everyone else is down. In return, when a member is revived, everyone gets back that 25%, plus a team-wide full heal and maybe even a short damage reduction buff (-50% for 5 seconds or so) so that the resurrected player can pick up the pace. This would actually encourage players to rez each other through multiple layers of incentives.
You can even combine it with the current system to add a further layer of risk vs reward element: you still need to collect resurrection charges, but they drop from the drones all the time. When a player goes down, you can choose how many charges you want to use to revive them. Use five charges, and they get back to the game, but not at full health, and the rest of the team doesn't get a healing charge either. Use ten, and everyone gets healed to full. Use 15, and everyone gets healed to full and get their energy bar filled too.
Alternatively, DE could do away with the resurrection charges altogether, and instead make it time based: when a player goes down, they cannot be resurrected for 3 minutes. Each enemy killed by the team reduces this timer by 1 second. Alternatively, when a player goes down, the team has to kill 25 enemies to rez them. Then for the second rez, it would be 50 enemies. Then for the next it's 75, and so on, meaning that death in penalized, but if the team does well, the penalty can be overcome. Or there could be special challenges that have to be done in order to rez the player, like killing X number of enemies with heavy melee attacks, or killing X enemies with headshots. Or you can mix and match the ideas, like earning resurrection charges by doing challenges instead of killing drones.
Really, there are dozens of ways DE could make this better.