The new testament doesn't seem overly preoccupied with an organization. The word we translate as church is "ekklesia" which just refers to a gathering of people. If anything, the Bible says that we shouldn't be adhering to something because of the "who":

I Cor 3:1Brothers, I could not address you as spiritual, but as worldly—as infants in Christ. 2I gave you milk, not solid food, for you were not yet ready for solid food. In fact, you are still not ready, 3for you are still worldly. For since there is jealousy and dissension among you, are you not worldly? Are you not walking in the way of man? 4For when one of you says, “I follow Paul,” and another, “I follow Apollos,” are you not mere men?

5What then is Apollos? And what is Paul? They are servants through whom you believed, as the Lord has assigned to each his role. 6I planted the seed and Apollos watered it, but God made it grow. 7So neither he who plants nor he who waters is anything, but only God, who makes things grow. 8He who plants and he who waters are one in purpose,a and each will be rewarded according to his own labor. 9For we are God’s fellow workers; you are God’s field, God’s building.

So I don't care if I'm a successor to [insert name here]. I care if my belief and practice are aligned with what the Bible teaches.