You should be following the 1 white path before the n black paths, not afterwards. In particular, following "3 blacks then 1 white" in the last step, when the final digit is three, is an error; it will produce the wrong modulus for any number that is not divisible by 7.

    (1) 1 white then 1 black,  ending up in state 1
    (6) 1 white then 6 blacks, ending up in state 2
    (0) 1 white then 0 blacks, ending up in state 6
    (3) 1 white then 3 blacks, ending up in state 0

The graph keeps track of ((the number so far) mod 7); following a white arrow multiplies your total by 10, and following a black arrow increases your total by 1.