You're right about my background. Still, I know many experimental PhD students who enjoyed great freedom in deciding what to work on. It's not always possible

You're right about my background. Still, I know many experimental PhD students who enjoyed great freedom in deciding what to work on. It's not always possible in the most expensive experiments -- a student probably can't easily repurpose the LHC for their own ends. But outside those huge experiments there's often a lot of flexibility, provided (a big caveat!) the PI is on board, and the student has ideas of their own.