House-elves can be ‘carers’, but we don’t know how typical it is.
When Barty Crouch Jr. is broken out of Azkaban, the Crouches’ house-elf Winky takes care of him. She nurses him back to health, and then takes care of him after that as well.
“That grave is empty. The house-elf nursed me back to health. Then I had to be concealed. I had to be controlled.”
- Harry Potter and the Goblet of Fire, (Chapter 35)
Winky was his carer, with him all the time, and was responsible for tending to him.
“I was always with the house-elf. She was my keeper and carer. She pitied me. She persuaded my father to give me occasional treats. Rewards for my good behaviour.”
- Harry Potter and the Goblet of Fire, (Chapter 35)
Barty Crouch Jr. was an adult at the time, not a child, and an unusual case because he was supposed to be in Azkaban, but this does show that house-elves can be used as carers. It’s unclear whether most pure-bloods have their house-elves tend to any children, but it’s possible.