Circular Problem with Equity/Entity (wacc) approach with private and public companies?

Hi,

At first about a public company:
1)Do I always know the equity value of a public company? (equity is 100% shares and therefore value is market capitalization)
2)Or is it possible a public company has only lets say 30% shares (market value visible) and 70% different owners (value not clear)?

If 1) is true then I never have a circular problem with public companies.
Then I only have the circular problem with private companies.

Now about private companies:
3)Do I sometimes know the equity value for a private company? (and therefore have no circular problem)
4)When I don't know the equity value we don't know the beta for the company precisely enough (we could estimate it with peer group beta) and therefore have the circular problem, right? Or is there another reason for the circular problem (is it possible there is even when we know the beta and equity value the circular problem)?
5) There is different options how to get the beta for a private company, why don't we use one of those?
Resulting in no circular problem (assuming the beta and equity value is the only reason for the circular problem)..

Thanks,
Kai