The common feature between Jeep Wranglers and Land-rovers of the type mentioned in the original post (along with a few others known for similar behaviour) is the solid front axle. A disruption on one side affects the other side. Sloppiness in anything affects both sides.
I'm not aware of decent independent suspension designs getting into positive-feedback steering situations. Yes, they can have "vibration" due to a wheel being off balance or a tyre worn unevenly and they can be "sloppy" if the bushings are shot, but I've never seen one go into what us motorcyclists would call a "tank-slapper" - a nasty positive-feedback steering oscillation in which something impacts the steering and kicks it (let's say) left, then the restoring-to-center forces kick it even harder to the right, then the restoring-to-center forces kick it even harder to the left, with the steering wheel oscillating hard in ever-increasing magnitude until something in the steering mechanism reaches a mechanical limit that contains it.
I've never owned such a vehicle ... I've been in a couple of them for long enough to hate the way they ride. Every bump gives a sideways ride-motion kick to the whole vehicle, and I can understand why. The front geometric roll centre is significantly above ground level, which means a one-wheel bump pulls the contact patch of both tyres sideways relative to the body as the axle tilts as it goes over the one-wheel bump - which in reality means the body is getting a sideways kick in addition to the bump motion. I've seen quite a number of lift-kit-equipped 4x4 vehicles in which the front track-rod and steering arm are significantly angled downward from the vicinity of the steering box (chassis end) to the axle end, which means in addition to the inherent sideways kick from the roll centre being up high, there's another asymmetrical sideways kick from the track-rod actually physically moving the axle sideways relative to the body and doing so differently when going over a one-wheel bump on the right side compared to a one-wheel bump on the left side. Evidently the people who drive such vehicles, don't seem to care. It won't be me ...