You are describing the classic DW aka death wobble. It is invariably caused by a loose suspension component. It can be made worse by lack of castor or negative toe in, but if everything is tight, lack of castor and toe in just makes it skittish. Many alignment techs will just grab a joint and try to wiggle it, and if they can't move it, they call it good. If the joint is under tension, you simply can't always wiggle it. Rocking the wheel back and forth applies the forces in alternating directions, causing parts to move.
Finding the loose components is fairly easy, but impossible for one person by themselves. For it to oscillate, there has to be something that allows the movement to start. Place a friend in the vehicle and have them rock the wheel back and forth by slightly more than the play in the steering wheel. That normally means about 15 degrees. While they are doing that, climb under and carefully place your hand on each joint, letting your hand bridge from one side to the other. In other words on a TRE, place your hand on the knuckle and over the TRE. If the TRE is loose as the person rocks the wheel back and forth, you will feel the play. Repeat this for all of the steering joints. This works sometimes on ball joints, but not all the time. In that case you sometimes have to lift the vehicle by the axle and take some or all of the load off of the tire, which will allow the joint to move without restriction. Also look/feel the control arm bushings. It may be harder to see them or feel them move, but they are a common source of instability.
Steering stabilizers do not fix the root cause of DW, they only retune the rate of oscillation to a less common frequency so it doesn't ever get started. They do nothing to get rid of the base problem. There are a lot of voodoo fixes and ideas out there about THE ANSWER to DW's. Don't just throw parts at the problem.
A vehicle CAN be aligned with bad ball joints. It just won't hold an alignment since the knuckle can move around.
