Etale-analytic comparison without elementary fibrations

A theorem due to Artin says that for a smooth scheme $ X $ of finite type on $ mathbb {C} $ and a locally consistent buildable sheaf $ F $ we have an isomorphism
$$
H ^ * _ {and} (X, F) approximately H ^ * (X ( mathbb {C}), F)
$$

where the LHS is the ├ętale cohomology and the RHS is the cohomology of the functors derived from $ F $ in analytical topology. One of the proofs I know is that there is open coverage of $ X $ by K $ ( pi, 1) $ the spaces. However, the appearance of K $ ( pi, 1) $feels a little odd to me. Is there any alternative proof that does not rely on K $ ( pi, 1) $ neighborhoods?