So your answer is no?
If so I disagree. Regardless of whether you use Schwarzschild or Ingoing (Eddington-Finkelstein) coordinates, the tilting is what you see when you draw light "cones" on a t vs. r (or t* vs. r*) plot. It's a coordinate effect.
Since coordinates are nothing physical, and you can't ever speak of shape or orientation of a light cone from the observer's perspective, why make a big deal of the tilting at all?
As for time travel, that occurs when the tilting is such that one's future light cone intersects one's past light cone. It isn't an issue in a black hole (the future cone is directed at the singularity, closed timelike loops don't become possible), it only means that from our perspective the gravity of a black hole sucks things away too strongly for their rockets or even their radio messages to approach us.