No, it is really not.

You left out the "accelerate forward" bit. That changes your reference frame. Without changing your reference frame, you don't get FTL to turn into time travel.

If you have FTL but on a time scale determined by one reference frame, then as I pointed out in https://news.ycombinator.com/item?id=37012594, no causality issues exist. You need FTL in 2 reference frames to create a closed time-like loop.

Does the machine have to be the source of the acceleration, or would ~any gravitational force count?

When you bring in gravity, you bring in general relativity. And now the phrase "at the same time as" becomes an entirely arbitrary choice of coordinate systems.

So it becomes complicated. But as long as FTL goes forward in time by some unique global definition of time, there is no causality problem.