Unless you live in a very isolated place, there are lots more options than public school or homeschooling. I'd check out this great unschooling blog yes-i-can-write.blogspot.com
, if you're interested. I was particularly interested to read about "free schools", which are schools that kids are enrolled in, but where they only have to "attend" as much as they want, and they decide their own interests to pursue. The kids also have equal decision-making power as the rest of the staff, and can be as much or as little involved in the community aspect of running the school (including conflict resolution) as they wish.
Other than that, I think the best (healthiest) option for kids is unschooling. Many people think that unschooling is only possible for parents with lots of time and a good network of support - the blog also talks about why that isn't so.
I don't think the negative impact of public schools on children can possibly be overstated. What you are seeing in terms of what your kids are learning from school about power relations is a total case in point - the wetiko disease that infects modern culture centers on the abuse of power (giving up our power to authority by learning to be "obedient" to those who dominate us, and in turn dominating those lower down on the hierarchy) - and the lack of empathy that this requires.
Unfortunately, given the sheer amount of time that kids spend in school compared to the rest of their waking life, I just don't believe that any amount of healthy parenting can truly counteract the negative influence of the enculturation that happens in public school. Its fighting not only an uphill battle, but a losing one. Not to say that many people don't fight the enculturation within themselves (usually the kids who reject school in one way or another), or learn to undo its damage later in life, because many (some) do - and devoted parents providing a healthy example to follow and a healthy home life does matter a lot. But that doesn't change the fact that public school damages kids no matter how much they choose to reject its influence (speaking from experience - I may very well be stronger for it, but I'll carry the scars for life).