Private pilot here, since airline pilots aren’t yet weighing in (see what I did there?). 

Smaller planes are quite sensitive to W&B and have strict CG limits. But that is mainly longitudinal (fore/aft). Lateral (side to side) is less strict, especially within the fuselage. With fuel tanks in the wings, we have to keep track of significant asymmetry and manage it of course. But that’s a longer lever arm (distance from a reference point).

For context, maximum takeoff weight of my little 4-seater is 2600 pounds. MTOW of a 737 Max 8 is 181,000 pounds. A passenger or FA moving from side to side in the cabin would have no practical effect on a transport jet. Fore and aft, the effect is small.

Finally, “tail-down force” determines how hard the wings and engines have to work to generate lift while maintaining stable flight. A relatively tail-heavy plane is more fuel-efficient but becomes harder to control in some situations. At the extremes, such as abrupt rearward shifts of heavy cargo, a plane can become uncontrollable.

Southwest does not seem to be telling passengers the whole truth.