The need to track an FTL target in realtime is the origin of the plot hole. It creates a reliance on sensor devices that are accepted as being omniscient, whose limitations depend entirely on the needs of plot or whatever random plot devices can be introduced to create those limitations.
As I've said many times, it would be better for everyone if treknology worked within a set of fixed limitations. "Can't beam with shields up" is a really good example, especially since it's applied more or less consistently to good dramatic purpose. "Can't fire while cloaked" is another one, although the reason for it has become obscured over the years and it is now a bit of an absurdity. The same standard could -- and actually should -- be extended to all technologies, with their limitations sketched out ahead of time.
Limiting sensors to STL velocity would require writers to use them more realistically, at least to the point that you cannot instantly know what's happening at a distance and you have to think about how close the other ship/planet/station would have to be for you to really know what's going on over there. It might also add a bit of a detective element to some stories; if you arrive at a colony and discover it's been destroyed by someone, you can warp a shuttle or probe out to the edge of the light cone and watch the attack with a telescope.