I imagine some didn't care but I imagine some felt awkward or even resentful of the concept. I wonder, if that's the case, how they rebelled or protected their own kids. Was there any effort to formally overturn this policy? Did any places simply ignore the idea of the "one drop rule" and employ a more nuanced view of race?
Well, there are cases like that of Thomas Jefferson, who freed the children he had with Sally Hemings. Though Sally Hemings, herself the daughter of Thomas Jefferson's father-in-law, was freed neither by the father-in-law nor by Jefferson.
Then here's another anecdote: in 12 Years a Slave, the narrator, Solomon Northup, speaks of a woman, Eliza, and her two children who were kept with him in DC after he was enslaved. The children, or at least one of the children, were fathered by the white slave master. He lived in a house with Eliza and the two children. They were treated as free, but they were not free. He promised to free Eliza and his two children. His white legitimate daughter hated Eliza and her two half-siblings. One day, while he was in town, his white daughter told the slave and her two half-siblings that she was taking them into town to be freed. Excitedly, they dressed in their finest clothing and went with her joyfully. But actually she was selling them down the river. That's why, if you see the movie, Eliza is wearing a beautiful dress and the children are also dressed in finery.
So. Some were embarrassed by their enslaved children. Some didn't give a shit. Some felt a bond and at least promised them good things, though they didn't necessarily deliver. Jefferson took his own sweet time in freeing his children, he didn't do it immediately.