There were a number of problems - the start/end times for the tournament in Settings - Tournament - Dates & Deadlines were set to the same time, and the rounds on the schedule were actually set to occur on a different day than the day scheduled for start/end. There were also 2 room pools ("Rooms" was empty).
I removed and readded the site from the tournament, fixed the deadlines, then cleared the rooms out of the pool, added them back in, reassigned the rounds to that pool, and checked in the Round Settings from the schematic to make sure everything was set. It looks like it's placing rooms now.
I think several of these problems were caused by importing data from another tournament - that sometimes causes issues like this. Usually, clearing things out and redoing that part of the setup will fix the problem.