Articulation and keeping the wheels on the ground has its place. But, in my experience, the #1 reason a vehicle ceases to go forward is that it has bottomed out. This would be on one of the lowest hard surfaces of a rig. Mostly either the lower A-arms or the diff. How does air suspension help with this?
Having just recently taken my new (to me) LR3 off-roading in soupy North Carolina mud, I experienced this phenomenon firsthand and learned something useful: air suspension is superior to a solid axle in muddy ruts. The underbody clearance of my LR3 in off-road height (granted, I also have 265/65/18 tires, which are a little taller than stock) is pretty generous, somewhere in the neighborhood of 11 inches. In extended mode it's a little more. My LR3 sat in the ruts and straddled the ground in the middle no problem.
Behind me on the trail was a 1995 RRC LWB, lifted, with larger-than-stock tires. It got repeatedly stuck, once to the point where even traction mats were no good and I had to extract it with my tow strap. When you looked behind the RRC there was a trench in the ground carved out by the diffs. Sample size is one, but in this case - a literal side-by-side comparison - all the air suspension vehicles on that trip (including stock D5s and LR4s on street tires) easily outperformed a built RRC. Some of that is likely due to Terrain Response vs. the VCU B-W transfer case, but none of the independent suspension vehicles got stuck because they bottomed out. None of them got stuck at all.
About 100 feet further along from here is where I had to tow the RRC. Traction mats helped at this point but later it became too deep.
L405 RR on 20"(?) wheels and street tires - no problem.
(Edited to add photos)