Thanks for the welcome!
We leave the stumps until we decide an area should be cleared of all trees, and replanted. Pretty common for this area of the country.
When an area is going to be cleared, we cut any remaining trees, then use a skid loader and dig out all the stumps/roots. After they are removed we plow the area, before replanting.
We do grind the stumps every spring to speed up the rotting process, and make mowing a little easier.
A lot of the time we will plant a new tree close to a tree that is big enough to sell, so theoretically when the larger one gets cut/sold the newly planted one will have plenty of room to grow, and the stump from the cut tree will rot while the little guy grows.
We also plant beside or between stumps whenever possible.
We've always looking for a better way of doing things.