Is it a wise move to outsource project management?

Imagine a IT service provider or main contractor S who outsources the project management portion to company A, systems integration portion to company B, building an integrated solution from 4 different product companies- companies C, D, E and F.
You may ask so what would be the role of the IT service provider S then? Well it would be just for pure account management and escalation of critical issues to the customer.

The fact that once the project management portion is outsourced it would mean the control is taken away. In one of my projects, that is exactly what happened. Company A while managing the scope creep, risks, issues and schedule delays as best as they could, their lifeline would remain there as long as they keep S informed and seek direction from S.
A has a difficult time managing B, C, D, E and F since S is the one who fronts the monetary and contractual stuff with B, C, D, E and F. A tries to keep everything professional and raises to S for any issues. S underestimates the criticality of the issues and brush the issues aside resulting in an accumulation of tension in the inter-working relationships among the different companies.
S also underestimate and remain out of touch with the issues the customer faces with the project since S relies on A to front them.
At the end of it, the project suffers resulting in frustrations from the customer, delay in the progress and scope mismanagement.

Is it a wise move then for S to outsource the project management aspect? Not in my opinion, cause project management is not just a delivery nor an adminstrative function for S. Rather project management means to S the vehicle to materialize the planned payment milestones in each of the project phase gates. Project management is core function to S in my view, and that makes it illogical for S to outsource it.