Thanks to everyone for all the advice. It's helpful to hear from people who have done this sort of work before.
The ground is a very soft/sandy clay out in the pasture area (to the very rear of the pictures I posted). Once you get into the timber, it is the same stuff but loaded with organic matter and it all becomes even more soft and loose. As I walk through the timber, it feels like I am walking on one of those rubber/foam mats. There's a noticeable "give" to the ground. This probably explains the erosion issues I am having, as you can almost stick your hand into the ground and grab a handful of soil. It just washes away if there's no roots in it.
There is no running water back there. Just whatever drains off the pasture during rains.
My current pick-up is 2WD without a locking differential and can move around in the drier and flatter areas if I put snow chains on it and the ground is dry. If the ground is wet...forget about it. I just dig a hole and go nowhere. I've had to unstick myself using boards, come-alongs, and shovels. In some cases, by letting the ground dry out first. It's no fun at all. Right now, I park at the edge of the wood and fell all the trees in that direction. Otherwise, I get to carry the wood out by hand. That gets old in a hurry and stops me from doing anything with the wood in the interior.
Good point about being able to use the blade to clear a trail and push debris away. I hadn't considered that...goes to show how much I have to learn from the more experienced folks here.
I guess I am reading the tractor recommendations as being able to carry more weight and the truck recommendations as being able to do the work faster. Neither one of those are big factors for me, as I'm just one guy doing this as a weekend warrior. Doing one tree in a weekend would be plenty for me. The smaller footprint, lower parts cost, smaller fuel expense, etc. is what makes this mini-dozer appealing to me. I am (possibly over-) concerned about causing problems with erosion, so the idea of a truck or tractor putting big ruts into the ground bothers me.
Decisions, decisions....whatever way I end up going, I'll try to remember to come back here and update.
Also might take some pictures of the worst of the gullies, just for bragging rights.