In the end, FTL sensors have effectively become plot holes
In the end? Don't you mean in the beginning?
TOS never bothered with an impression that things would be happening in the vastness of "realistic" space. It would have been odd and distancing if the ships didn't behave like their WWII counterparts, with radar, radio and whatnot. Kirk's folks were quite aware of what was happening around them in real time, and the very few cases of communications delay (chiefly "Balance of Terror") were more like bureaucratic delays at the other end than physical limitations. If anything, the attempt to introduce delays-due-distance was a desperate one, detrimental to both drama and consistency.
If every ship were in effect cloaked while at warp, we'd be in an even worse jam trying to explain why wars aren't conducted by surgically destroying the opponent's defenseless homeworld during Day 1, Hour 1.