We do grind the stumps down to ground level, so mowing isn't an issue-like you, we don't let the grass/weeds run rampant. The big trees get mowed several times over the summer, the little trees get mowed once a month or more if we've had lots of rain.
There are approximately 50 acres of trees-so it's a lot of mowing!
I think one of the big factors is that we plant our seedlings directly in the field-we don't have an area for babies that will get transplanted later. I think that may be why you can dig the stumps more easily (that and your sandy soil). I'm guessing our trees have more/deeper roots than yours?
We plant 1500-2500 seedlings each year-so transplanting would be a huge job.
When we clear an area with the skid loader it looks like a mine field-huge holes left where the stumps were-the clay sticks to the roots like glue. When the stumps are gone, we till the area several times, then run a drag over it to hopefully get things leveled back out.
In the fields there are spots that it would be impossible to get the skid loader into without damaging the surrounding trees, maybe the longer reach of a backhoe would make a difference?
We would love to be able to remove all the stumps each year-we just haven't found an effective way to do it, especially with the larger trees/bigger stumps.