My name is Legion, for we are many.
I can make up some of reasons for that:
1. They COULD have hated Halsey, but she did give them the choice to leave or stay after training, as with Soren-066 in Halo Evolutions. So if they did choose to stay, they didn't have anybody but themselves to blame.
2. They were fed a lot of BS and propaganda at a very young age to effectively brainwash them. Modern day humans aren't much different, what with children serving in WW2 and present conflicts.
3. They were facing two wars at the same time, so they'd feel honored to be picked to be superheroes. Seeing as those little kids were physically and mentally proficient, as well as competitive in nature, they would want to fight.