Etale-analytic comparison without elementary fibrations.

A theorem due to Artin states that for a smooth scheme $ X $ of finite type about $ mathbb {C} $ and a constant locally constructible sheaf $ F $ we have an isomorphism
$$
H ^ * et (X, F) approx H ^ * (X ( mathbb {C}), F)
$$

where the LHS is ethnic cohomology and the RHS is cohomology functor derived from $ F $ In the analytical topology. One of the tests that I know comes from establishing that there is an open cover of $ X $ by $ K ( pi, 1) $ However, the appearance of $ K ( pi, 1) $I feel a little weird. Is there any alternative test that is not based on $ K ( pi, 1) $ neighborhoods