My question is related to a statement:
If a pendulum is experiencing free fall, then it will not oscillate.
The statement is true in the sense that its acceleration is (approaching to) zero, then according to the period equation $T$ would be infinity.
I think this sense comes from General Relativity. However, I'm not sure exactly where this result is coming from, and why we should consider General Relativity for this pendulum, rather than considering it to be a pure Newtonian scenario.
why we should consider general relativity for this pendulum, rather than a pure Newtonian senario
There is no need to consider general relativity at all for this scenario. Newtonian gravity clearly replicates this behavior.
In a Newtonian context we would consider a free falling frame to be non-inertial. As such there would be a fictitious force that is the same magnitude as gravity but opposite direction (but not a 3rd law pair). This force would cancel out gravity for no net force. Thus there would be no oscillation
Indeed you don't need GR here as was clarified by the previous answer. But if we look at the situation from the GR point of view, then according to the equivalence principle, a small enough non-rotating freely falling frame, which is falling for a short enough amount of time, is the closest thing we have to an inertial frame, i.e., a frame were the laws of SR hold. And this frame will not detect the external gravitational field, thus will see the pendulum as not oscillating.
