Would an organization whose sole purpose is to give money to charities (aka a Charity of Charities) qualify for a tax exempt status?

Yes.  It's called a non-operating charitable foundation. There are two types of 501(c)(3) charitable foundations in the USA: operating and non-operating.  An operating foundation uses its funds to directly further its charitable cause.  A non-operating foundation is limited to donating funds to other 501(c)(3) charities and governmental entities.  As noted, typical family foundations are non-operating.  So not only are non-operating foundations common, it is often easier to get initial IRS approval for one than for an operating foundation.

It should, if its express purpose is to fund other recognized nonprofits. It would still require its own nonprofit status to be approved by the IRS and its operations open for review.