Yes, they likely wouldn't and how many leaves can someone carry, even if you break off a branch with leaves on it and have someone like Ruby carry it, it still would only last a certain span of time, and in the movies as I recall none of the gang thinks about carrying leaves with them as I recall.
The one thing is if the quests is done for dinos outside the Great Valley they may allow your character to rest there and get water and food, if they are the non nomadic but resident types, and they may have some knowledge also. Farwalker, nomadic types may have some information about other places since it's unlikely a resident herd would have much first hand info about other then where they live. & a nomad herd may let you rest with them also.