Universality of the Galois action on the fundamental group of $\mathbb{P}^1\setminus\{0,1,\infty\}$

We prove that any semi-simple representation of the Galois group of a number field coming from geometry appears as a subquotient of the ring of regular functions on the pro-algebraic completion of the fundamental group of the projective line with $3$ punctures.
Comment: Corrected the proof of Lemma 6.2, other minor edits. Comments welcome!