SoundGuy, for the sake of discussion (different than 'argument'):
"without anything else securing the trailer in place there's no way I'd trust those Andersen "chocks" from absolutely preventing the trailer from shifting"
I only have a single-axle trailer, so I can't speak to best-practice for dual-axle trailers. But, I can see your point about not trusting the Andersen leveler, if you don't get the tires level with one another. But, the Andersens won't work to get the tires level with one another, unless you make a ramp for the lower tire to get level with the other tire, before the leveling process begins, correct?
So, my question is, when your trailer is being parked where the ground runs downhill front-to-back (or vice-versa), do you make provisions to have the tires on each side level to one another, either on one side, or on both sides? If so, there would be no downhill tendency for the tires, especially if you use the stabilizers and tongue jack. If not, then being over-zealous with chocking would be called for. But, it seems to me that, if you properly use the stabilizers and tongue jack, there would be less downhill pressure on the tires, so the chocks won't have that much work to do.
Do you try to get your tires level to one another (on the same side of the trailer) as part of your leveling process? Curious, since I don't have dual axles, and it seems to me that one would want to get them level.