Lack of lifts is not what's causing delays in today's skyscraper elevator systems. It's lack of shafts or "lanes" for the lifts. Today, the only way to have eight lifts serve a floor is to have eight shafts (*). But if the lifts could overtake, then it would suffice to have just two shafts, like a two-lane road: eight (or more) lifts could then zip up one lane, down another, and swerve to the opposite line if a stopped lift at some floor blocked the way up or down.
Now, even two shafts take up a lot of space. So the starship would probably just have one shaft, and then little side alcoves for overtaking - much like the railway network typically only has a single rail for long stretches, plus side rails at suitable intervals so that trains going in opposite directions can pass.
(*) Okay, there's another way, sort of. The situation with modern skyscrapers is actually so desperate that there are plans of stringing up several lift cabs from the same cable, atop each other, so that when people board at Level 8, others can disembark at Level 7. Current technology cannot create cable-free, sidestepping lifts yet, so vertical stacking of some sort or another is better than nothing.