Yes it assumes a single universe. I'm not aware of any multiverse solution to this. There is, however, the Novikov self-consistency problem which could solve this. It's entirely speculative, of course.

And allowing non-flat space time refines but doesn’t alter the picture.

For instance, it’s a common misconception that wormholes circumvent the relativistic prohibition against ftl travel. Actually, if a wormhole could facilitate travel between two events faster than a light ray moving between the same two events on the outside of the wormhole, then all the same causality troubles would ensue. This just highlights that that the particular mode of transport between events is not important to the argument. (Whether it be warp drive or secret tunnel.)