The annoying player that always asks if he can have a certain item is not caused by the influence mechanic. He would/will ask as many questions while playing first edition in regards to money. And if the GM just handwaves it and says: "Just subtract xxx thrones and have it" it will most certainly not be positive in the long run. So you still need to have GM controll over availability.
The influence system just substitutes the roll with keeping book and adding/subtracting thrones.
Prefer what you want, but it is not more or less work than before.