That your G is a test category is stated in the last sentence of 4.1.20 in the paper of Cisinski you mention. This case is also treated in more detail in section 8.3, where it is shown that the left Kan extension/restriction along both adjunctions induced by your "forgetful functor" are Quillen equivalences (8.3.8).
(By the way, if I recall correctly, it is a corollary that if we give simplicial sets the Joyal model structure instead, then both adjunctions are still Quillen pairs, and they realize the two adjoints to the inclusion of (∞,0)-categories in (∞,1)-categories.)
