I have straight turf tires front and rear and the only place on my lawn where I have damaged the lawn is where the front tires are when I'm hooking up or unhooking my 3 point implements. I have a quick hitch system, and unless that's lined up -just so-, then it's a problem to attach, so I often have to do lots of manuvering, sharp turns at very slow speeds. Given all that, I'd be more impressed if that place on my lawn didn't show signs of wear and tear.
So the short of it is, other then what I've described above, I don't seem to have problems with my 2920 using all around straight turf tires.
I always have my FEL attached. I regularly traverse my lawn with my tractor.
There are some precautions that I do take. I make sure I'm out of 4WD when hooking and unhooking and I don't tractor on the lawn when the lawn is wet.
It could just be that you're soil is just naturally softer then mine, or that you're using a different grass.
The only advise I can offer is to switch the front to all turf tires and see if that makes a difference. It's a rather expensive test though.
Larry