Every person experiences an Adrenaline rush if they are in a dangerous or combat-oriented situation ; therefore it would make sense for every person to have an adrenaline rush, in various situations - especially at the start of the game, where the Soon-to-be Survivor (probably) never met a Zombie - or giant monsters and wild, dangerous animals. I’m not sure how easy you could create such an adrenaline system though - it might work best if every monster (and npcs for that matter too) have a fixed ‘Danger’ Value ingame, and the Player has a danger-limit, a threshold. If the visible monsters/situation is above said Threshold, an adrenaline rush starts, improving your stats, reducing pain and improving stamina. Said Threshold would slowly rise, the longer the player survives, up to an upper limit, which would be the ‘finished’, battle-hardened Survivor, who fought hordes, jabberwocks, visited labs and so on.
This Threshold could also be used for stuff like ‘too scared to move’ - if the situation is double or triple your danger-limit you could be scared shitless, thinking ‘im dead’ and all that stuff, and simply not move for a round or two, before the survival instinct kicks back in. That would only be human. In return, a high danger-threshold could prove to be a light mental immunity against horrors and such.
The Adrenaline trait on the other hand could either play with the adrenaline threshold - or it could improve the boni you get from an Adrenaline rush.
However, as I said before - I have the feeling that this system would be very, very hard to implement, and it would take away from the roleplay experience and might be frustrating if your player stares blankly at 2-3 zombies for an hour, causing him to die.