You've cut danelions before, haven't you? Where does the next flower top end up?
Just below the blade-cutting height...right?
Question: You've got two boys?
Answer: One BIG sandbox.
Forget dandelion control w/o chemicals...I've tried...you just have to keep everyone off the lawn for two/three days (what the bag/bottle says to do), then TA! DA! - No yellow dots in the back yard...
Otherwise, live with the yellow dots or the backyard becomes a rock garden/sandbox...
With little boys, I'd stay away from the rocks for a couple of years yet!