I don't think that the demonstrator is giving the contraption a shove.
Suggesting that the demonstrator gives a shove is not necessary.
Let me call the direction in which the four wheels are pointing 'forward' and 'backward'.
The contraption has a bottom plate (to which the four wheels are attached) and an attached vertical plate. Then there is a movable vertical plate, the connection allows the movable plate to swing from side to side. Finally, a movable arm is connected to the movable vertical plate.
I'm spelling out all of this because I need names. From here on I will refer to 'the movable plate', and 'the movable arm'.
I will refer to the freedom of motion of the the movable arm as motion with respect to the movable plate. That is, even if the movable plate is swung sideways to 90 degrees, I will still refer to the motion of the movable arm as up/down.
Let me first discuss what the demonstrator would have to set up if the gyro wheel is not spinning.
Then he can simply lift the movable arm, and when he releases the movable arm the contraption as a whole will start to move, simply because the weight at the end of the movable arm is swinging down.
Now to the actual demonstration, with a spinning gyro wheel.
In the starting position the demonstrator turns the movable plate sideways, but the movable arm isn't moved with respect to the movable plate.
The demonstrator releases the contraption.
The gyro wheel is in a high position so it starts to fall down.
If the contraption as whole would not be on wheels but on an air table then the contraption would not be in a position to exchange momentum with the outside world. On an air table the contraption would mainly shift from side to side, with its center of mass remaining in the same position.
On the wheels: since the contraption as a whole won't move sideways the swing of the movable plate is relatively larger.
As the movable plate swings down the axis of the spinning gyro wheel is forced to change orientation. That forced change of orientation causes the movable arm to swing up (gyroscopic precession). As the movable arm swings down again the contraption as a whole is dragged forward by the weight of the down-moving arm.
By the looks of it:
I do see that the downswing of the movable plate overshoots the lowest point. But I think by that time the movable arm is back to resting against the movable plate. From that point on the only direction the movable arm can swing is away from the movable plate, or back to resting against the movable plate.
In the demonstration you see that for each run the system is freshly prepared.
The movable plate is lifted sideways, then the contraption is released.
The case is really not different from what you would see if the gyro wheel is not spinning. It's just that if the gyro wheel is not spinning you go straight to lifting the movable arm.
When the gyro wheel is spinning you add the intermediate step of lifting the movable plate sideways, and then the gyroscopic precession lifts the movable arm.
A closer look at the case of the contraption suspended on an air table:
Even if the contraption is suspended on an air table it would still exchange momentum with the outside world when the sideways lifted movable plate is released. When the movable plate is released, and it starts to swing, the air suspension needs to exert a net torque to keep the bottom plate of the contraption in the horizontal orientation. So that is still exchange of momentum with the outside world.