The first thing that comes to mind is whether that approach would be feasible at all, and the answer is that it would not:
struct X : enable_shared_from_this {};
std::shared_ptr<X> p( new X );
std::weak_ptr<X> w( p );
p.reset();                      // this deletes the object
if ( w.use_count() ) {          // this needs access to the count object
                                //    but it is gone! Undefined Behavior
If the count is stored in the object, then no weak_ptr can outlive the object, which is a breach in the contract. The whole idea of weak_ptr is that they can outlive the object (if the last shared_ptr goes out of scope, the object is deleted even if there are weak_ptr)