If you think about it, a support is just an applied force or moment, right? If you have a simply supported beam with a force $F$ in the midspan, the supports will be two forces, each $F/2$ in the opposite direction.
But then, what if, instead of supports, you simply had two forces $F/2$ applied at the ends? That'd be exactly the same, right?
The difference between supports and forces is that supports are "flexible": they generate the forces needed to keep a structure stable. An equivalent force is only valid for a specific loading case.
So sure, if you have $F=100$ and apply two forces at the ends, each equal to $S=50$, that'll be identical to having supports at those ends. But if you then have $F=200$, those applied forces won't cut it. Meanwhile, if you have supports, well, then they'll generate forces of $S = 100$ each.
Basically, think in terms of order of operations: your structure suffers some loading configuration $q$, and then from those you calculate your supports. From a procedural perspective, supports are different from forces in that supports are a function of forces.
In your case, if you apply a bending moment at the hinge (it'd have to be on both sides of the hinge), then sure, you could make a balanced structure for a specific loading criteria. But you'd just be mimicking the behavior you'd get if you simply didn't have a hinge at all, and simply had a continuous (though bumpy) beam.
So, what you need isn't a bending moment at the hinge. What you need is to get rid of the hinge.