As an addition, the autothrust does not engage when the levers are moved to FLX or TOGA; it arms. Moving the thrust levers to CLB only will do nothing with the autothrust if it is disengaged.
Once autothrust is armed, moving the thrust levers back to the CLB detend can engage it (subject to some conditions).
On a visual approach, moving the thrust levers to the CLB detend for max climb thrust, will not engage or arm the autothrust.