Parents also wish the same. Many want their kids to be independent. The problem is identifying at what age the kid has to start their hygiene practices on their own. I honestly say, it varies from each kid. Parents should understand their kids ability before letting them to do their chores on their own. Because, brushing and bathing is basic to one's cleanliness. Letting the kids to do them on their own and their lack of perfection may lead to other health problems.
So I personally feel, kids can be allowed to brush and bath from 4 or 5 years, but supervision is must till they attain 6 years of age.