The sprout from the rhizomatous root system. Anything that kills the sprouts other than like fire or removal, will kill or injure the mature plants. If they bother you too much, you should remove all.

In spite of the claim that treating the sprouts would harm the mature mother plant, I use this method routinely when needed (not with sassafras specifically, but with many other woody plants) and have never had a problem. (My claim presumes the sprouts are small and that you do not have a full thick lawn of them! If the sprouts are not small, make them small before treating.) While this may sound risky, consider that for glyphosate to have its usual intended detrimental effect, a certain minimum amount of it must be taken in by the plants system. In treating these small sprouts not anywhere close to that amount of Glyphosate is applied. If considering this method caused anxiety or great apprehension, one could treat a small sample of sprouts and verify the results for themselves before proceeding to treat the whole lot. Really, though, I have been doing this for ever.

Obviously injecting poison into the toes will affect the head. Due to much greater mass, the poison may be diluted and just produce minor distortion of the main stem growing point.

I suppose it's like our anti-biotics (literal translation: against life), you treat enough to kill the problem and hope to not kill the whole organism. Roll the dice!

I have the same issue, and have resorted to Round-Up, using Ken's "undiluted drops on the freshly-cut stump" method and it does get rid of that particular sprout without affecting the mother tree in any visible way. I do not use that on sprouts right by the mother tree (less than 2 feet away), but all the other ones - in the lawn, 20 feet away, nestled in the peonies....). If the mother trees are well-established (mine is at least 40 years old), they will survive.

Try to keep them, if you can; besides being native, they are beautiful in the fall, loved by native bees, and a larval food for butterflies. Besides, they're cool,.with the different leaf shapes.