There is such a thing as incomplete broodiness, where a hen will display broody behaviors to a degree, but not be fully broody and committed. I have one like that, she's really confusing (and confused).
The thing is, chickens are SO far removed from what nature intended, that they've lost a lot of their instincts along the way. They have been extremely heavily selected by people for qualities that people want out of them - maximum number of eggs, maximum meat, pretty colors, etc. And any time you have heavy artificial selection pressure like this, you lose something along the way. Chickens' reproduction is completely unnatural at this point. What other bird lays eggs continuously like that? Normal, wild birds have a breeding season, lay a handful of eggs, raise the chicks, and are done for the year. But we have twisted nature into giving us an egg pretty much every day, year-round. And that's not the only way we've broken chicken reproduction. A lot of hens will never go broody in their lives. In nature, their lineage would get wiped out. There's no such thing as not reproducing in nature. Broody hens will often kill their chicks for no reason. Again, not something you'd see in nature, to this degree. Hens will also kill other hens' chicks raised within the flock - within their own social group. No sane social animal does this. The point being - chicken reproduction is all kinds of F-ed up, by people's selection processes, and cannot be relied on. Some hens go broody and do a good job sitting and raising chicks, but a whole lot fail all along the spectrum from sitting to raising. Some aren't committed and will abandon the eggs. Some will move to a different nest. Others will sit on some days, and forget the eggs on others. Some will sit during the day, but not at night. Etc. etc. etc. Having a good, reliable broody is priceless, because so many of them can suck in so many different ways. So using a new broody is always a gamble. Yours sounds like she's not a good choice, so don't give her fertilized eggs to sit on. Break her if she's getting too carried away. If there's no danger of her getting malnourished or overheated, then you can leave her to her ambivalence.