A couple of things may be going on. First, the root system may be compromised from being restricted to the container - there may be circling or girdling roots present that prevent the tree from sending roots out laterally to anchor and establish stability. These must be corrected for the long term viability of the tree. Second, is how the tree is staked. A single stake allows too much flexibility/movement in other directions. A triangle of three stakes outlining the perimeter of the planting hole will provide the maximum support. Make sure the attachment is at no more than 2/3's the height of the tree and there is some flexibility or give in that material. You do want the tree to flex and move with wind, etc. - that's what encourages trunk development. And the staking should be present for only the first year......if you need to stake longer than that, I'd bet money on root problems.

The problem now is that the tree doesn't have a stable vertical position within its root ball. It's almost as if whatever roots it had for stability were broken, which may have happened during one of the spring storms here (we've had a few storms with wicked strong winds).

On a still day (no wind), I'm sure I can find a way to balance the tree in a vertical position, but any wind at all and the trunk shifts at its base... meaning that the only way to hold it still is to stake it tightly... unfortunately, if I stake it tightly, it will never learn to stabilize itself... so this seems like a catch 22.

Doubtless it is root-bound, this is very common and often very severe. You must correct the roots if the tree is going to have a secure future.

When I planted it last fall, I sliced the root ball into four sections and splayed it open... Root bound though it may be, I think the problem in this case is that the trunk of the tree is only "loosely" attached to the root ball, which is to say that it has a lot of freedom of movement if you push the trunk this way or that.. the root ball is not moving, but the trunk is.

If the trunk is that loose but the rootball remains stationary, then it may well be that major roots were severed, either courtesy of wind storms or perhaps your rabbit or other critters. That doesn't spell a good outcome for the tree.

Not an ideal time to do so, but I'm not sure you've got a lot to lose at this point. I'd dig up the tree, remove ALL the soil from the rootball (gently, using water) and examine the root structure. If there is still a decent root system (a couple of major, intact, non-circling structural roots and sufficient finer feeder roots, then go ahead and replant. I'd also consider staking again, as per above, especially if subject to a lot of wind. If not an adequate root system or if obviously broken or torn major roots, you may want to think about starting over with something new.

FWIW, washing the roots or essentially bare rooting the tree may be the way to go in the future with any new plantings. This way you can be assured that any issues with being root bound or correcting girdling or circling roots have been properly addressed.