so if they don't have medical supplies ready there's a chance the second player would immediately die and continue to not get to play? Yeah no, that's a horrible mechanic
If they don’t have medical supplies ready I don’t think the first player is surviving long anyway, so the second player wouldn’t need to wait long. Also this game isn’t really known for letting you die immediately. Maybe rather then a timer it could be some kind of parkour/inventory management/etc. minigame the second player has to win to “finish training” and get sent to rejoin the first player right after, then it’s skill-based on both parts.