Do you ever wonder why it is so much easier to squat deep using a load than it is with just bodyweight?
The reason is because of relative stiffness. By loading the torso with a bar, as in the front or back squat, the body naturally reacts by stiffening the torso to protect the spine. The load now makes the torso stiffer than the hips, and the body always follows the path of least resistance.
To test this out, do a bodyweight squat when you are feeling "cold" and your hips feel tight. Notice how difficult it is to drop into a deep squat. Next try holding some front and side bridges (pillars, planks, whatever you want to call them) for about 30-40 seconds each. Now try the bodyweight squat again. Did you notice a difference? If you did the bridges correctly, it should have felt much easier to sit into a deep squat. Basically all you did was increase the neural signal to the torso muscles, increasing their activity, thus "stiffening" the torso, making the hips less relatively stiff than the torso.
If somebody seems to have a hip mobility issue, immediately check the stability of their torso. More often than not, their hips have become too stable, probably from sitting too much, making up for the instability at the torso.
Physics always rules.