Homotopy theory based on an other space than the circle

Question

This question might seem philosohical. To me homotopy theory is based on two piece of information in the category of topological spaces : the circle $X = S^1$, and the reduced suspension functor $F = \Sigma$, then everything can be deduced from there : the homotopy groups functors are $[F^n X,-]$, CW-complexes are defined using homotopy quotient of spheres $F^n X$ etc. Is there a reason why no other homotopy theories are used, for example, for $X = RP^n$, or $X = K(\mathbb{Z},n)$ etc. Is is possible to have a homotopy theory also by changing the functor $F$ ? Or the only sane choice is the right adjoint to the mapping space functor $Map(X,-)$ ?