Why can't we just bud?

Sarah is at that age where she's pondering periods a lot.  Today, out of nowhere, she said, "I wish we could reproduce by budding instead of having periods. A tiny head would sprout on top of your shoulder and grow until a little clone of yourself got big enough to drop off and be your baby. Do all animals have periods?"  I've never thought about it before and didn't know.  She googled it and found that most mammals reabsorb the endometrium rather than shedding it. She wants to know why we can't just reabsorb.  I don't know.  I'd like to be a reabsorber myself.

In completely unrelated news...Noah came into the kitchen and spotted the spinach I just bought last night.  He excitedly squawked, "Oooooh, we have spinach!" then was immediately embarrassed and said, "I'm not good at being a kid. Kids should not get excited about spinach."