You assume that the massive amount of energy propelled in front of you ever stops. It doesn't. It's just a matter of what ends up being nuked; if you're lucky, you can aim it right at a black hole and never worry about it. If you're unlucky, you just wiped out an entire planet.
Nobody's done a profile of what such an energy blast would look like, at great distance, and compared it to what we've seen of Gamma Ray Bursts, have they?
(Let's say that many distant galaxies have civilisations with such drive systems. They've all, independently, realised the dangers of their methods if they don't 'aim off', away from any vulnerable world in their own galaxy, so each segment of any flight is aimed to head for 'dark sky' with nothing but extragalactic bodies (perhaps not even fellow galactic local cluster members), and even then ideally at somewhere that only the Hubble Deep Field would spot anything in. Eventually, however, the dissipated and extremely acute conical section intersects
somewhere. Like us. ...or else it has been used as an actual weapon, against some 'nearby' world, and we (again) just happen to be on the same ultimate trajectory, only mildly occulted by the intented target, but still saved from the same fate through sheer distance.)
However, probably not. But its implementation might be a good basis for a story (short or otherwise). A sudden cessation of GRBs could indicate that distant civilisations are now
not aiming at us (because they now have us mapped, and are avoiding even subjecting us to the extremely low-level side-effects). Although, theoretically, their 'supralightspeed' GRB-creating drives would let them get here (intentionally aiming 'out', into further empty sky, as much as they are able)
before their purely light-speed GRBs stop (or even start?) arriving. Depending on operational limits like cool-down times and exactly
how much past lightspeed they can travel.