First of all, don't use Epsom salts. They are magnesium sulfate and are only useful to add magnesium to soil that is lacking the element. There are seemingly millions of places that recommend them for everything. They are only scientifically proven to help when more magnesium is needed. Most Epsom salts get washed away and go into our water system. Too much magnesium can keep plants from taking up calcium and this woudl harm the plants you have around the stump. See this for more info: https://extension.umn.edu/manage-soil-nutrients/coffee-grounds-eggshells-epsom-salts#:~:text=Adding%20Epsom%20salts%20to%20soil,water%20that%20percolates%20through%20soil.

Nearly all apple trees grown today are grafted. The size of the tree is dependent on the root stock used. So, if a grower took three cuttings from one Haraldson tree and put each one on a different variety of root stock, one would be a dwarf tree, one a semi-dwarf and one a standard. Most apples would not be able to grow well without being grafted to better, stronger root stock. So, you could cut off one of these shoots and graft it to another apple tree, but trying to root it will not work. Trees also don't grow roots from a cut branch like many houseplants. They have a very different vascular system. 

You should look to see where the shoots are coming from. If they are below the graft (or you cut the tree off below the graft), the new growth is coming from the root stock. That usually means not great tasting apples. The trees that are used for root stock have robust roots, but pretty lousy fruit. If the shoots are coming from above the graft, you could let one grow and get a new tree. If course, it will eventually be too big again. But, maybe your garden will have changed by that time so you are fine with a larger tree. 

Remember that the plants that were growing in the shade below the tree may be in full sun now. 

You could rent a stump grinder to be done with this right away. Otherwise, if you are vigilant in cutting off all of the shoots, eventually the roots will weaken and die. 

Hope this helps. 

Thank you!