Normally we say that nobody “owns” a public charity.  Although a person or persons control the charity, nobody “owns” it like people own Facebook. Nobody receives dividends from its profits or makes money if the entity is sold. Charities and other nonprofits often have members, with rights and responsibilities akin to shareholders of a business, but we don’t normally talk about the members “owning” the organization.

You are correct that a public charity often serves as the sole member of another public charity. This is a frequent model in healthcare systems and other multi-organizational group of entities providing coordinated services. I don’t recall ever hearing about a private foundation serving as the sole member of a public charity, but I don’t know any reason why it could not do so. The IRS determines public charity status on the basis of the activity of the organization or its public support.  (See Ready Reference Page: “Calculating Public Support Percentage”) Public charity status is not determined by who has control of the organization.

The one area in which we do talk about “ownership” of a charitable organization is where the controlled entity is a limited liability company. State laws normally talk about the owners of an LLC since most LLCs are formed for profit. Where a limited liability company has only a single member, the LLC is normally considered a “disregarded entity” for federal tax purposes. For federal tax purposes, all of the activities of the LLC are considered those of its sole member. Therefore, if the LLC is “owned” solely by a public charity, it is treated for tax purposes, although not for state liability purposes, as a public charity. If the LLC were owned solely by a private foundation, it would normally be considered for tax purposes as a private foundation so that, in that situation, the private foundation would “own” another private foundation, not a public charity.