I have to agree with the statement that a skid steer would not be the way to go. You want something that WILL tear up your lawn, buy a skid steer. The name itself will tell you that to skid to steer means something is not moving while something IS moving and if it's turning with a part of it stationary, then it's going to tear up whatever is under the stationary part. That's how skid steers drive, and they really leave some tracks, and yes, they would not be good on slopes.
Honestly, the first things that came to mind were Kubota's BX23 and Powertrac's system. I understand the power tracs are pretty slope-hugging, and have all manner of attachments available that are supposed to be easily interchanged. The only thing about them is that you must buy direct from the MFG, and have service done by someone in your area, or send back to PT for the service. A tractor will usually have many dealers in many locations.
The BX, I think, is pretty good on slopes, but we'll have to hear from an owner. They are lower to the ground than many others, but I don't remember if the stance is wide enough for a steep slope.
I rather agree that your slope may be more accesible than a 45 degree angle would permit. 20% would be my top, but I must confess, I'm not sure I would recognize for sure the degree until I was starting onto it, and my "pucker factor" would say, "Get the heck off this slope!"
Massey also makes a good sub-cut (like Kubota) that may also serve you well enough, but once again, I'm not sure of slope fitness of either.
A kubota BX with backhoe and loader probably would be a good choice.
Come on owners, tell us about slopes on your BX.
John