Is there any part of a flight where the pilot would pitch down?
The most common case to pitch down is to speed up, not to reduce altitude, for example when transitioning from a climb (at climb speed) to level flight (accelerating to cruise speed). You can transition to level flight by reducing thrust, but that'll leave you level at climb speed, and is usually not what you want.

Changing configuration will usually require pitching changes to maintain stable flight. In particular during the approach-to-landing, extending the flaps will substantially increase the drag, and in order to maintain airspeed, you'll need to pitch the nose down. If you don't, you'll bleed off airspeed and potentially eventually stall unless you overpower the drag with thrust.

There are of course unusual circumstances that will require a pitch-down as well, obstacle avoidance for example (e.g. airplanes). As mentioned in the comments, to make a speedy descent you'll probably also want to pitch down (such as in case of pressurization problems or fire on board).