Yes, but onky minimally, as a side effect. The purpose of throtle stick is not to change the vertical direction... For that there is an other dedicated stick...
By the way, how does that work in a real helicopter? Have never try...
In a non-turbine helicopter you have, primarily 3 controls.
1. Collective and Throttle (throttle is like a motorcycle) - Left Hand Pull up for increased pitch and turn hand control to right to increase throttle
2. Stick (typically right hand)
3. Rotor (peddles at your feat - Left & Right)
Collective increases/decreases the pitch of the rotor blade and makes the craft go up, level or down
Throttle controls the RPM of the engine (and the rotor)
Stick is direction control (in all directions) Push the stick forward and you move forward, pull back and you go in reverse and left right are as expected.
Peddle also control yaw, and left peddle must be increased if throttle is increased to maintain heading or control yaw caused by torque of engine.
The actual control of the aircraft is done through the use of all of these controls, simultaneously. Hovering is a joy to learn, especially close to the ground.