It's often done, often with good results. But you may have some issues to deal with:
1. If you don't grind the stump thoroughly and deeply, it will be physically difficult to put the new tree in exactly the same place.
2. If you do use a stump grinder, you'll have a fluffy mixture of wood chips, soil, and air, that may take a couple years to settle down on the site, and in the mean time, that rotting wood will use a lot of nitrogen, as bacteria compost it. You would therefore likely need to increase the proportion of N in your fertilizer program for the first few years. The tree will tell you that, with less-than-deep-green leaves.
3. I suppose if you can physically pull the trees, rather than cut and grind, you may avoid the above problems.
4. Disease -- in some parts of the country, there may be phytophthora or armillaria infecting the original trees, to which the new tree may be susceptible. But if your original trees are healthy, that will likely not be a problem.
5. It may be wise to have the soil tested, even if you pull up the old trees so the ground-up roots are not a problem, in that the old tree may have removed most of some specific nutrient(s), and would temporarily alter the recommended fertilizer regime for the new tree.