It depends. The state I live in forces all home schooling to be done through a religous institution. There are a number of churches that will gladly take your money to sanction your homeschooling, but there is no option for someone who doesn't want to pay that tax to a church to home school. I'll grant you that a lot of home schoolers are doing it to avoid evolution and whatever other things they disagree with. But I can tell you that all of the public high schools where I live are complete shit except the "Magnet" schools which are extremely competitive, focused on specific disciplines, and will give you the boot for a single non excellent grade or disciplinary action. So far as private schools go the same is true in my observations they are all tied directly to some religous institution or another. In the end if any one of my children doesn't seem to be able to hack it at the Magnet schools I'll probably relocate to anywhere else that doesn't have a completely broken school system.
My kids are vacinated for most everything they can be. But even if you look at measles, which is the current panic disease, the mortality rates are a joke. Something like 1 in a thousand cases in the years before vaccination was mortal. With modern health care I would expect it to be even less. The risk that non-vacinated kids pose is laughable at best, and it's not like vaccination results in 100% immunity anyways. If you're immune compromised, that sucks, but let's face it you are incredibly likely to die from something like the flu anyways. The massive amounts of public panic over that risk going up a few percentage points is completely unwarranted in my opinion.