A robot closes distance fast
A global planner can take real time, tens or even hundreds of milliseconds, to search for a good long-term route, because it only needs to run occasionally and the environment it's reasoning about doesn't change much in that window. The local obstacle avoidance layer doesn't have that luxury. A robot moving at a realistic speed, even a modest indoor speed of around 1 meter per second, can close the distance to a newly detected obstacle in just a few seconds, and at higher speeds that window shrinks fast.
This means local avoidance typically has to run many times per second, often somewhere in the range of 10 to 20 hertz or more, just to keep its picture of the world current enough to react in time. An algorithm that produces a theoretically excellent avoidance maneuver but takes half a second to compute it is worse than useless in this context, because by the time the answer arrives, the obstacle has moved and the robot has moved, and the computed maneuver no longer matches reality.