Sir actually the source of confusion here was that the omega already has some physical significance of its own...its not an arbitrary variable...so u see this gives a wrong impression
but sir this is something we know that shm is a one dimensional motion, so i think it makes sense to write F=-kx..since it was observed experimentally that the force is varying linearly with displacement..
just check this out ...the solution of the differential equaltion came in the form of k/m...but it is only after the assumption that k/m=w^2 that we can see the w term in shm general equation