Okay, so the trajectory object component has to be applied to the thrown object prefab, so that's solved.
However, my grenade is not exploding though, and I get an error about item use:
NullReferenceException: Object reference not set to an instance of an object
Opsive.UltimateCharacterController.Items.Actions.GrenadeItem.StartItemUse () (at Assets/Opsive/UltimateCharacterController/Scripts/Items/Actions/GrenadeItem.cs:44)
Opsive.UltimateCharacterController.Character.Abilities.Items.Use.AbilityStarted () (at Assets/Opsive/UltimateCharacterController/Scripts/Character/Abilities/Items/Use.cs:457)
Opsive.UltimateCharacterController.Character.Abilities.Ability.StartAbility (System.Int32 index) (at Assets/Opsive/UltimateCharacterController/Scripts/Character/Abilities/Ability.cs:434)
Opsive.UltimateCharacterController.Character.UltimateCharacterLocomotion.TryStartAbility (Opsive.UltimateCharacterController.Character.Abilities.Items.ItemAbility itemAbility, System.Boolean ignoreCanStartCheck) (at Assets/Opsive/UltimateCharacterController/Scripts/Character/UltimateCharacterLocomotion.cs:1245)
Opsive.UltimateCharacterController.Character.UltimateCharacterLocomotion.TryStartAbility (Opsive.UltimateCharacterController.Character.Abilities.Ability ability, System.Boolean ignorePriority, System.Boolean ignoreCanStartCheck) (at Assets/Opsive/UltimateCharacterController/Scripts/Character/UltimateCharacterLocomotion.cs:1037)
Opsive.UltimateCharacterController.Character.UltimateCharacterLocomotion.TryStartAbility (Opsive.UltimateCharacterController.Character.Abilities.Ability ability, System.Boolean ignorePriority) (at Assets/Opsive/UltimateCharacterController/Scripts/Character/UltimateCharacterLocomotion.cs:1023)
Opsive.UltimateCharacterController.Character.UltimateCharacterLocomotion.TryStartAbility (Opsive.UltimateCharacterController.Character.Abilities.Ability ability) (at Assets/Opsive/UltimateCharacterController/Scripts/Character/UltimateCharacterLocomotion.cs:1012)
Opsive.UltimateCharacterController.Character.UltimateCharacterLocomotionHandler.TryStartAbility (Opsive.UltimateCharacterController.Character.Abilities.Ability ability) (at Assets/Opsive/UltimateCharacterController/Scripts/Character/UltimateCharacterLocomotionHandler.cs:231)
Opsive.UltimateCharacterController.Character.UltimateCharacterLocomotionHandler.UpdateAbilityInput (Opsive.UltimateCharacterController.Character.Abilities.Ability[] abilities) (at Assets/Opsive/UltimateCharacterController/Scripts/Character/UltimateCharacterLocomotionHandler.cs:128)
Opsive.UltimateCharacterController.Character.UltimateCharacterLocomotionHandler.UpdateAbilityInput () (at Assets/Opsive/UltimateCharacterController/Scripts/Character/UltimateCharacterLocomotionHandler.cs:99)
Opsive.UltimateCharacterController.Character.UltimateCharacterLocomotionHandler.Update () (at Assets/Opsive/UltimateCharacterController/Scripts/Character/UltimateCharacterLocomotionHandler.cs:73)
It's probably a small mistake I've overlooked somewhere, but where?
EDIT: Actually, it seems the Trajectory Object goes on the item pickup prefab (not sure why yet), and the grenade script goes on the thrown object
(seems kinda obvious now ?).