What known methods are there to approximate an arbitrary homotopy type by an algbreo-geometric object in a concrete (read: computable) way?

I know this in an ambitious question, so maybe I should narrow it to a special case: what known methods are there to approximate $K(\pi,1)$ spaces (i.e. spaces with non vanishing homotopy groups only in degree one) by algrbreo-geometric objects?

I am familiar with Toen's schematic homotopy type and J.Pridham's pro-algebraic homotopy type, but it seems all but impossible to directly compute nontrivial examples for interesting homotopy types. I am not familiar with any concrete examples of either of these constructions in the literature, does anyone have a reference containing examples?

Perhaps another question tangential to the question above is the following: is there a notion of an algebra-geometric Eilenberg-Maclane space $K(G,1)$ for an arbitrary non-abelian group $G$?

Finally, is anyone familiar with any other methods in the literature, perhaps using pro-schemes?

What methods are there to approximate an arbitrary homotopy type by an algebraic-geometric object in a concrete (read: computable) way?

I know this is an ambitious question, so maybe I should narrow it to a special case: what methods are there to approximate $K(\pi,1)$ spaces (i.e., spaces with non-vanishing homotopy groups only in degree one) by algrbraic-geometric objects?

I am familiar with Toen's schematic homotopy type and J.Pridham's pro-algebraic homotopy type; however, it seems all but impossible to directly compute nontrivial examples for interesting homotopy types. I am not familiar with any concrete examples of either of these constructions in the literature. Does anyone have a reference containing examples?

Perhaps another question tangential to the question above is the following: is there a notion of an algebra-geometric Eilenberg-Maclane space $K(G,1)$ for an arbitrary non-abelian group $G$?

Finally, is anyone familiar with any other methods in the literature, perhaps using pro-schemes?