As often as Simon tells them they should fire the person helping then pick songs, that wouldn't make much sense. Once they choose their songs, I can understand secrecy so as not to ruin the show.
I have a feeling the list was super short and may have not benefitted all of them. jmo
They have a vocal coach so I am sure he or she has a lot of input on picks. I would say the pick comes down to what they like and how comfortable they think they are with the song material and how big a risk they are willing to take at each step of the competition. IIRC in an iterview the producers said they all the get the same list to choose from. So depending on the song catalog it can be pretty big say for category Country or Rock week or small if they are choosing from one artist.