My broodies were the exact same way.. Neither EVER left the nest to eat/drink/poo. I had to do it manually. I've had one that would poop if I didn't move her in time.. I will never forget the smell.
Yes. I learned this lesson the hard - er, smelly - way too. My broody was not getting up to eat or drink (I could tell by the level of the food) so I started bringing her food but not making her go far from the nest. Bad idea. This is why we have chicks that are a week apart in age. She had somehow rolled the eggs in the poop. They were completely covered. I ended up washing the ones I knew were developing and putting them in a newly purchased incubator. Got mama some new eggs and took her off the next every morning. I'd take her about 8 feet from the nest and a light bulb would go on: Eat, Drink, Poop, Dustbath! Back to the eggs in 5 minutes. After two weeks of this she got the idea and started taking herself off the nest once a day to go about her business.