No. All the equation tells you is that the total energy momentum (as expressed by the matter tensor and the gravity pseudotensor) is conserved. There is no total flux out or in. It still can be that in some regions it's in positive and others out positive, and as long as those balance out you get a total flux of zero.
It is conceptually no different than your other example with two hypersurfaces.
And of course it still depends on the pseudotensors which are not covariant, so little physical sense. If you had some preferred coordinate such as what you get with a timelike Killing vector then it's ok, even if you only have that at infinity where you integrate, or I would think for any similarly asymptotic flat compact spacetime (I am not sure if one can make this mathematically correct because not clear what asymptotic to hypersurface not at infinity would be. You might be able to prove this using some conformal transformation).
But it certainly does not mean that the energy is zero.
You might know that if one talks about lightlike infinity being asymptotically flat, the energy momentum conserved is different than if you use spacelike hypersurfaces. In the spacelike case it is the ADM mass (which for gravitational radiation emitted by a black hole is the mass of both the black hole plus the radiation), whereas for lightlike infinity it is only the black hole mass, as the flux of energy through those lightlike hypersurfaces takes the gravitational radiation energy out. That's the so called Bondi mass, and the black hole looses it as it radiates.
I don't have the Weinberg book to know or interpret exactly what he said, but the treatments for the kinds of mass (or energy in the sense that they are the same) that are conserved is also treated very well, both physically and mathematically, in Wald.
So, it gets tricky, but unless you define the total energy momentum as T-G (which you can trivially get from a Lagrangian variational treatment) you don't get zero in general, and you only have pseudotensors.