I have to disagree with you there. If they were only limited to certain choices the show would have been over a long time ago because that would have been rigging. Someone would have found a lawyer went after AI and they would have talked by now.
The way I understand it, there’s a “pre-cleared” list of songs each week. Contestants are free to go off that list, as long as they stay within the theme (and a few contestants have pushed the boundaries there over the years), but there’s no guarantee that they will be able to secure clearance in time for broadcast.
This happened to a few contestants that I can recall — once someone was on their fifth choice because they didn;t receive clearance on the first four. And then there was the flap with Archuleta last year where he “mixed in” part of a non-cleared song for his performance, which cost 19E some extra bucks in royalty payments.