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Weight and weight balance is a big deal for aircraft stability. For example, a bunch of weight in the back of the plane will cause it try to pitch up more, which can matter quite a bit when landing.
Don't they have sticks for that up front?
I think it can be compensated for to some small degree. But... A deeper explanation is needed.
A plane has a center of mass (or gravity) which is the gravity balance point of the plane. It also has a center of lift which is like the center of gravity but for the lift force of the wing. See pic:
If the center of lift and center of gravity aren't in the same horizontal location along the length of the plane, the plane has a net force that rotates the nose up or down.
It's like if you had a yard stick and balanced it on your finger. And then added a weight somewhere between your finger and one end. The yard stick will start to rotate (tilt).
The horizontal stabilizer applies a force as well, and it is trimmed to balance the other forces and set the angle of attack (pitch) of the plane. The elevator allows momentary adjustment of pitch.
I may not describe this perfectly but... When a plane has too much weight towards the rear, it will be unstable (and dangerous) to fly. When a plane has too much weight towards the nose it will be somewhat sluggish but stable.
Balancing nose heavy is preferable to tail heavy in general due to the stability issue. A little bit of tail heavy is a bit dangerous and a bit of nose heavy is no big deal. I think another factor is that the tendency of a plane to pitch up could result in aerodynamic stall (loss or lift). Of course, if a plane is too nose-heavy, you can't climb.
Front to back balance is really important to how the plane handles. Side to side balance matters but in terms of passengers / cargo, they're more narrowly distributed about the center of the plane compared to their distribution front to back.
https://www.boldmethod.com/learn-to-fly/performance/what-effect-does-center-of-gravity-have-on-your-airplane-performance/
Oh man I was just trying to make a pithy comment, now you have me actually learning stuff?
Other problem is that a weight shift has to be countered with elevator trim, which can increase aerodynamic drag and therefore fuel consumption.