Let's say a helicopter wants to move forward. To do so, cyclic pitch is applied to the blades 90 degrees off which applies a torque on the blades to tilt the craft forward. Once the helicopter is tilted there is now lift upwards and lift forward. But why doesn't the helicopter just keep tilting forward, doing flips?
My first thought was that the pilot only applies the torque until they are satisfied with the tilt angle then returns to normal control. But this can't be the case since in many videos I've seen the swash plate remains tilted throughout the entirety of the movement. I can't think of another way that the helicopter could remain at the desired forward angle and not just flip repeatedly.
Very good observation - helicopters actually do want to flip over, in the hover. While hovering, the helicopter is in an unstable equilibrium and the pilot will have to make constant cyclic inputs to prevent the heli from flipping indeed.
Your observation of constant forward cyclic deflection is when flying forward, and this reveals the stabilising factor: airspeed.
The incoming frontal horizontal airflow component wants to blow the rotor backwards, which would actually happen with a centred cyclic. Maintaining forward cyclic prevents this from happening.
But the required forward cyclic input is very small, still requiring close attention from the pilot. In order to create an equilibrium state requiring a larger forward cyclic deflection, helicopters have stabilisers at the tail. With a nose-down tilted fuselage, forward velocity creates a nose-up moment.
The helicopter fuselage follows the tilt of the rotor plane, moving the CoG backwards, as depicted below. This creates a stabilising moment nose-up.