No, your homeowners/umbrella/whatever will NOT cover you for an employee. Those policies are structured to cover visitors to your home, occasional workers (like a plumber who should have his own worker's comp, etc).
Without it, your choice may be to either bankrupt yourself paying someone else's long term medical bills (imagine, say, a fall causing major bone damage while chasing your child), or being the asshole who fights your employee tooth and nail to prevent having to pay for those bills, or the even bigger asshole who just acts like you don't know it's part of your liability if it happens.
In any case, you are required by law to carry it, at least in MD.