New 5th wheel doesn't tow well

It's not a true GN pin. If it was it would connect on the base of the trucks bed. He would have an after market pin with a GN attachment. That would give him a 90* angle from the pin down to the GN ball in the truck bed. That could create the push pull.
But that's not what the OP has. His connection is in the same place as any other 5th hitch.

In terms of components you are correct. They are not exactly the same. And of course, the true goose neck has the vertical column attached to the fifth wheel and the ball is accessed at the truck bed. In the Anderson, the hitch is attached to the ball in the truck bed (the model that does not attach to the frame) and the rest of the hitch is just a platform to raise the second ball to the place where the fifth wheel modified ball receiver/coupler is attached to the existing pin. I never said they were identical, I said they function in a similar way to a goose neck hitch. And even Anderson on their webpages describe the Ultimate hitch as a 'Goose Neck' fifth wheel hitch. In many ways it functions like a goose neck as it can be hitched/unhitched on uneven terrain easily; it can be hitched/unhitched at angles that would preclude a fifth wheel hitch working properly; there is no plate, pin or jaws to hold the pin; and it is very light compared to most fifth wheel hitches. As to chucking, there probably is little difference since chucking is derived from the road surface, the trailer's wheels going over bumps, and the fact that in some cases with soft suspensions the fifth tries to lower the rear of the truck slightly when the trailer wheels ride over bumps. In the OP's experience, some suspension upgrades might help but probably not totally eliminate chucking.