The video below is a company propaganda video of course ... But i find it to be very accurate based on my use. I have never been able to get my ASV stuck, and I've used it in all sorts of muddy, wet-clay conditions.
And it's not just the traction that makes a CTL way better than a CUT for most of these types of conditions. They have a smaller overall footprint, are more maneuverable, can run much heavier-duty front equipment , substantially more hydraulic flow, and the lifting force of the loader blows away the tractor. My CUT, which has a great loader for the class (2500 lb rating) is easily outclassed by my CTL, with a 5500 lb tipping load.
I am slowly replacing my tractor implements with front hydraulic implements, as the CTL is just so much better for most of my (commercial) applications. I now have accumulated: standard bucket, stump bucket, oversized bucket, 4-in-1 tooth bucket, grapple bucket, Front Harley rake, Rockhound, 4-way Dozer blade, Tree Puller... so useful...
Regarding a CTL on grass... It all depends on how skilled of an operator you are and how careful you want to be. I can either destroy grass and turn it all into topsoil again, or I can barely leave an impression. It just depends on the work I have to do and how careful I am trying to be.
The ASV undercarriage design has the benefits of better traction, better ride, higher ground clearance, and lower ground pressure. The downside is that the undercarriage can be more expensive to maintain and replace the components than the other standard CTL platforms. I'm also actively looking to get another CTL, a larger 100 horsepower unit with a typical CTL undercarriage. That one will be used for larger grading projects and rock work, which takes a heavy toll on the ASV undercarriage system.
Anyhow, here's the vid.
ASV Posi-Track Loader 80HP CTL Shooutout - YouTube