The positive mass conjecture was proved by Schoen and Yau and later reproved by Witten. Total mass in a gravitating system must be positive except in the case of flat Minkowski space, where energy is zero. Since QG is intended to be a theory of interaction with force particles called gravitons, one may begin to wonder if the interactions are in fact the important defining features of the space in question. So does a theory with interactions also require that space be curved?
Dear Humble, because non-gravitational yet interacting field theories such as QCD or the Standard Model exist and they don't predict a curved space, the answer to your question is clearly No, interactions don't imply that the spacetime has to be curved.
However, the curved spacetime follows from many other assumptions - or combinations of assumptions - for example from the requirement that the gravitational force (respecting the equivalence principle) simultaneously exists with the relativistic Lorentz invariance.
The theorems you mentioned clearly assumed that the spacetime is allowed to become curved.
So far, I assumed that you agree that the existence of interactions is a property of a theory, not a property of a configuration. But what about the possibility that you meant the "existence of interactions" to be a property of a state, or a configuration?
Because the positive-energy theorem implies that the energy is strictly positive with the single exception of an empty Minkowski space, it follows that if you also agree that the Minkowski space "has no interactions in it", then every state that has interactions "in it" has nonzero energy and consequently has to lead to a curved spacetime.