How long can I leave shoots keep coming up where the tree used to be before it becomes dangerous?

Whether it can still be saved

The tree is already gone; the goal is stopping the regrowth. The reliable method is a cut-stump herbicide application — triclopyr or a concentrated glyphosate product painted onto the fresh cambium ring within minutes of the cut, ideally in late summer or fall when the plant is moving resources downward and carries the chemical into the roots. Once the cut face has dried and sealed, that window has closed and you are left with repeated foliar treatment of the sprouts over a season or two, or exhausting the roots by cutting persistently for several years. Do not train a stump sprout into a replacement tree: sprouts attach to the outside of the old stem and hold poorly once they gain weight.