Even though they arrived with a bit of time to spare, if they didn't want to arrive in the middle of the meeting they'd have to move soon. And if they wanted to claim that they were just here for their studies it would be even more important to either catch the people responsible before the meeting.. or wait until it was over, since the weird stuff supposedly happened afterwards anyway. Those meetings relied on a safe space and even without much information the Shamans could see how complete outsiders who didn't share the suffering could be frowned upon.
Then again: all it took sometimes was a nice, convincing pitch.
Not completely shooting down your ideas, just putting out a possible concern.
Also, if you want to get in before the meeting starts you can have two more rounds of discussion.