Those tires actually aren't terrible. They are some off-brand 30" with semi-aggressive tread. Not what I would have bought, but they came on the rig.
Normally, I don't need to air down when I'm on the trails. They are easy enough to navigate... when you actually stay on them. That was my mistake, backing up and turning. The hill you have to come down to even get back to that spot is much, much nastier than where I got stuck, a sandy wash that gets rutted out occasionally. I have no problems getting up that, even with fully aired up tires.
But yeah, once I get this erroneous thing cleared up with the government, my build will include some 33" MTs, a 4" lift, a new rear axle with an e-locker, and eventually (hopefully at the same time) a 350tbi and transmission. Down the line I may put an e-locker in the front as well.
Before any of that though, I'm going to pick up a hi-lift and some straps.