I don't think the implication was that a non-Tesla party would be faster at getting a Tesla part; rather they would be faster at getting a non-Tesla part.
Can dealership do that? Every time I went in for Nissan service, they emphasized the genuine parts (I suppose to justify the mark-up).
If a random mechanic can get a random non-Tesla part, why is there a need to affiliate themselves with Tesla? Bunch of shops in my area advertise that they "fix German cars" without flying a Mercedes-Benz or BMW logo.
I don't think the implication was that a non-Tesla party would be faster at getting a Tesla part; rather they would be faster at getting a non-Tesla part.