Because they are ridiculously incompetent. The fact these two managed to get on the tie fighter is a testament that it goes beyond being arrogant to just straight up wanting to fail. Cameras exist in the sequel trilogy. But I guess they don't believe in escorting high risk individuals.
It's not about forseeing that one loose soldier will cost them Starkiller base. It's as simple as letting a mentally compromised soldier wander free is a big risk. He could have just gone to the reactor and killed them all. He could have shot leadership. Fucking with people's minds tends to fuck them up.