Under normal conditions, the bike WANTS to be stable. Normal conditions are both wheels on the ground with good contact patch and a bit of rake angle on the front forks. IF the bike encounters something in the road like a surface irregularity or something on the surface of the road, the front end might deflect. Again, under NORMAL conditions, the bike will self correct and any kind of wobble will normally damp out.
Adding weight to the back of the bike affects the rake angle on the front of the bike. Normally, you would associate the greater rake with more stability, like say on a cruiser. However, cruisers are still balanced front to back so that the front end still has the correct weight on the tire to maintain the desired contact patch size. Simply adding weight is not going to cause wobbles, especially if the rear suspension sag is adjusted. Where the weight is added is what matters most. Adding weight BEHIND the rear axle lightens the front end. So even though you are increasing the rake, the lack of contact patch size becomes more of an issue, reducing the bike's normal ability self correct.
When you have a bike loaded with luggage that is high and rearward, you are lowering the inherent stability of the bike. Adjusting the rear suspension to maintain proper sag helps. Add too much weight behind the axle and adjusting your rear sag will not overcome the problem. The bike might seem fine in a straight line on a relatively smooth surface, but it will be much more sensitive to disruptions because its normal ability to self correct has been reduced. IF the rider is paying attention, the bike might feel a little more "nervous", which is a good indicator that you need to make a change! If the rider does not, then something as simple as an undulation in the road surface or even getting on the gas too hard in a curve can upset the bike and induce a wobble. Rider reaction can make it better or worse. Most people will instinctively react by either chopping the throttle, hitting the brakes, or both. THAT can cause a crash even when the wobble might have damped out had none of that happened. The reactions CAUSE a wobble to become a tank slapper.
The accepted means of correcting a wobble is to get weight on the front tire WITHOUT making sudden speed changes or sudden braking inputs. This is the reason for the advice to lay on the gas tank and get your weight forward. This increases the contact patch and restores the normal tendency of the bike to self correct, hopefully in time to prevent a crash.
Tire pressures, head bearings, and suspension settings can all affect the handling of the bike and contribute to wobbles. However, from what I have experienced and learned from reading about it, weight distribution is the biggest culprit, unless you have something REALLY extreme regarding the condition of the tires, bearings, and suspension. Those riders you see with MASSIVE piles of luggage on the back of their bikes are taking a BIG risk!
I have ridden with passengers and/or luggage and experienced the front end getting uncomfortably light. On hard acceleration, from a stop or out of a corner, there have been times where the front end came up and when it came back down, I got a wobble. In every case, I was able to ride through it because I didn't panic and make a sudden input to the bike. In most cases, doing NOTHING actually allowed the problem to self correct. In a few cases, I pulled myself forward to get more weight on the front end and that took care of the problem. Once I was aware of the issue, If I could not stop to correct it, I simply had to change my riding to be much more aware of the difference in handling.
I had one high speed wobble on my KLR and all that saved me there was the hand of God... I kid you not
I later figured out that causes were VERY loose head bearings, VERY soft springs in the front (which actually reduced the rake), and the nasty buffeting of an 18 wheeler I was approaching from behind. I truly thought I was going down and was already looking into the grass median to see if there was anything I'd hit once I parted from the bike. Then... just like that...it stopped... I had been gradually rolling off the gas and was shifting my weight back in anticipation of trying to jump clear of the bike as it went down. I had unwittingly reacted appropriately. Once home, I corrected the suspension/bearing issues and the bike never had a problem again.