Well, QFT can be also be safely done on a curved spacetime background, but you are right, you don't get dynamic gravitational effects from it. What I implicitly assumed is QFT+semiclassical GR, where one uses semiclassical probability-weighted stress-energy tensor as a source.
If that were true, MWI would have inter-world gravitational interactions. But it happens to be obviously wrong.
This is a thread where people can ask questions that they would ordinarily feel embarrassed for not knowing the answer to. The previous thread is at close to 500 comments.