The reduced thrust period around max-Q in the shuttle SRBs was called the "thrust bucket".
I'm old enough that I still got a lump in my throat every time I heard the Capcom call out "go at throttle up" at the end of the thrust bucket during a shuttle launch.
My understanding is that solid rocket boosters are still popular because they can provide a lot of thrust (as opposed to specific impulse) fairly cheaply. High thrust is important early in a launch when gravity losses are greatest (because the launcher is still pointed mostly up) and the stack is still pretty heavy. I still don't like to see them launching humans, though.
Gravity losses decrease as the launcher pitches down and mass quickly declines to only a small fraction of that at liftoff. High specific impulse then becomes more important than high thrust as the launcher builds most of its velocity. This is why high performance liquid fueled engines (especially those burning H2) are popular in upper stages, though there are many exceptions.