The Statute of Liberty is a pretty smart piece of architecture, but it has limitations.
To get to the observation level in the crown, which is over 250 feet above the ground level, there are a pair of spiral staircases. One is used by people going up, the other by people going down. It’s a tough climb, but it’s slow and, lets face it, the Crown is a very small area so you can’t stay up there very long.
But the only path up to the torch is essentially a ladder. I found a picture online
Now, there’s probably more room on the torch than there is in the crown, but the ladder is crowded and dangerous. There’s just enough room on the spiral staircases for two streams of traffic, and to get to the ladder, you have to take the staircase. Having people waiting on the staircase to get on the ladder would block the staircase as there’s not room for two people abreast.
The staircase is strictly one at a time.
As such, if someone wants access to the torch (and they do let people up there from time to time) they have to shut down the staircases and limit the number of people in the structure so they can all travel up the ladder at the same time and down the ladder at a different time.