I think you need to ask your three year old.
For the bulk of the camping I've done with my kids, I can't tell you how grateful I was to have brought along a set of kiddie snap together foam pads to protect the underside of our tents. It's also far more comfortable. Nowadays, I use Reflectix under the tents.
Ground "cloths" have always been more about protection, than waterproofness, but sadly this gets lost in translation. If you don't believe the extra protection is warranted, and are fine with the occasional hole, then don't lug it along.
I do agree that I have seen more miss-use of them than anything else, and people seem to forget that the tiniest amount of exposure will run the risk of allowing water get inbetween the floor and the groundcloth. In that situation, water will travel the path of least resistance, typically up through the floor of the tent.
Furthermore, you want to truly test the durability of a tent, just leave it up to anyone aged ten and under. The first thing my kids always did was jump up and down on anything squishy- like a $100 t-rest.
And no matter what, have fun camping with the little one!